Box plot
Section 4.3
Percentiles and quartiles
Section 4.3
Median
Section 4.1
Chi-squared goodness-offit test
Section 15.1
z-test and estimator
of a proportion
Section 12.3
Pie chart
Section 2.2
Bar chart
Section 2.2
Frequency distribution
Section 2.2
Chi-squared test and
estimator of a variance
Section 12.2
t-test and estimator of a mean
Section 12.1
Box plot
Section 4.3
Chi-squared test of a contingency
table
Section 15.2
z-test and estimator of the
difference between two proportions
Section 13.5
F-test and estimator of ratio of two
variances
Section 13.4
Range, variance, and
standard deviation
Section
ion 4.2
Percentiles and quartiles
Section 4.3
t-test and estimator of mean
difference
Section 13.3
Mean, median, and mode
Section 4.1
Line chart
Section 3.2
Stem-and-leaf
Section 3.1
Unequal-variances t-test and
estimator of the difference between
two means: independent samples
Section 13.1
Equal-variances t-test and estimator
of the difference between two
means: independent samples
Section 13.1
Histogram
Section 3.1
Ogive
Section 3.1
Compare Two
Populations
Describe a
Population
Problem Objectives
Covariance
Section 4.4
LSD multiple comparison
method
Section 14.2
Chi-squared test of a
contingency table
Section 15.2
Two-factor analysis of variance
Section 14.5
Two-way analysis of variance
Section 14.4
Chi-squared test of a contingency
table
Section 15.2
Simple linear regression and
correlation
Chapter 16
Least squares line
Section 4.4
Coefficient of determination
Section 4.4
Coefficient of correlation
Section 4.4
Scatter diagram
Section 3.3
One-way analysis of variance
Section 14.1
Tukey’s multiple comparison
method
Section 14.2
Analyze Relationship
between Two
Variables
Compare Two or
More Populations
A GUIDE TO STATISTICAL TECHNIQUES
Not covered
Not covered
Multiple regression
Chapter 17
Analyze Relationship
among Two or More
Variables
7:03 PM
Interval
Nominal
11/22/10
Ordinal
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DATA T YPES
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AMERICAN NATIONAL ELECTION SURVEY AND GENERAL
SOCIAL SURVEY EXERCISES
Chapter
2
3
ANES
3.62–3.67
3.68–3.71
4.37–4.38
4.58–4.60
4.86
12.51–12.53
12.116–12.123
13.42–13.44
13.73
13.123–13.125
A13.27–A13.30
14.27–14.32
14.47–14.50
14.66–14.67
A14.23–A14.25
15.17
15.43–15.46
A15.25–A15.28
16.45–16.49
16.73–16.76
A16.27–A16.28
17.21–17.22
A17.27–A17.28
4
12
13
14
15
16
17
Page
82
82
117
126
144
413
435
472
488
512
524
543
553
563
594
604
615
631
665
671
689
713
734
GSS
2.34–2.37
3.25–3.28
Page
31
64
4.39
4.61–4.62
4.84–4.85
12.46–12.50
12.103–12.115
13.38–13.41
13.70–13.72
13.113–13.122
A13.18–A13.126
14.21–14.26
14.43–14.46
117
126
144
413
434
472
488
511
523
542
552
A14.19–A14.22
15.18–15.21
15.39–15.42
A15.17–A15.24
16.50–16.53
16.77–16.80
A16.17–A16.26
17.16–17.20
A17.17–A17.26
594
604
614
630
666
671
689
712
733
APPLICATION SECTIONS
Section 4.5
(Optional) Application in Professional Sports Management: Determinants of the Number
of Wins in a Baseball Season (illustrating an application of the least squares method and
correlation) 144
Section 4.6
(Optional) Application in Finance: Market Model (illustrating using a least squares lines and
coefficient of determination to estimate a stock’s market-related risk and its firm-specific risk) 147
Section 7.3
(Optional) Application in Finance: Portfolio Diversification and Asset Allocation (illustrating
the laws of expected value and variance and covariance) 237
Section 12.4
(Optional) Application in Marketing: Market Segmentation (using inference about a proportion
to estimate the size of a market segment) 435
Section 14.6
(Optional) Application in Operations Management: Finding and Reducing Variation (using
analysis of variance to actively experiment to find sources of variation) 578
APPLICATION SUBSECTION
Section 6.4
(Optional) Application in Medicine and Medical Insurance: Medical Screening (using Bayes’s
Law to calculate probabilities after a screening test) 199
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Statistics
FOR MANAGEMENT AND ECONOMICS ABBREVIATED
9e
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Statistics
FOR MANAGEMENT AND ECONOMICS ABBREVIATED
9e
GERALD KELLER
Wilfred Laurier University
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
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Statistics for Management and Economics
Abbreviated, Ninth Edition
Gerald Keller
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Jack W. Calhoun
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Brief CONTENTS
1
What is Statistics? 1
2
Graphical Descriptive Techniques I 11
3
Graphical Descriptive Techniques II 43
4
Numerical Descriptive Techniques 97
5
Data Collection and Sampling 161
6
Probability 175
7
Random Variables and Discrete Probability Distributions 217
8
Continuous Probability Distributions 263
9
Sampling Distributions 307
10
Introduction to Estimation 335
11
Introduction to Hypothesis Testing 360
12
Inference about a Population 398
13
Inference about Comparing Two Populations 448
14
Analysis of Variance 525
15
Chi-Squared Tests 596
16
Simple Linear Regression and Correlation 633
17
Multiple Regression 692
Appendix A
Data File Sample Statistics A-1
Appendix B
Tables B-1
Appendix C
Answers to Selected Even-Numbered Exercises C-1
Index I-1
vii
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CONTENTS
1
What Is Statistics? 1
1.1
1.2
1.3
1.4
2
Graphical Descriptive Techniques I 11
2.1
2.2
2.3
3
Introduction 12
Types of Data and Information 13
Describing a Set of Nominal Data 18
Describing the Relationship between Two Nominal
Variables and Comparing Two or More Nominal Data Sets 32
Graphical Descriptive Techniques II 43
3.1
3.2
3.3
3.4
4
Introduction 1
Key Statistical Concepts 5
Statistical Applications in Business 6
Large Real Data Sets 6
Statistics and the Computer 7
Introduction 44
Graphical Techniques to Describe a Set of Interval Data 44
Describing Time-Series Data 64
Describing the Relationship between Two Interval Variables 74
Art and Science of Graphical Presentations 82
Numerical Descriptive Techniques 97
4.1
4.2
4.3
4.4
4.5
Introduction 98
Sample Statistic or Population Parameter 98
Measures of Central Location 98
Measures of Variability 108
Measures of Relative Standing and Box Plots 117
Measures of Linear Relationship 126
(Optional) Applications in Professional Sports: Baseball 144
viii
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CONTENTS
4.6
4.7
4.8
5
Introduction 176
Assigning Probability to Events 176
Joint, Marginal, and Conditional Probability 180
Probability Rules and Trees 191
Bayes’s Law 199
Identifying the Correct Method 209
Random Variables and Discrete Probability Distributions 217
7.1
7.2
7.3
7.4
7.5
8
Introduction 162
Methods of Collecting Data 162
Sampling 165
Sampling Plans 167
Sampling and Nonsampling Errors 172
Probability 175
6.1
6.2
6.3
6.4
6.5
7
(Optional) Applications in Finance: Market Model 147
Comparing Graphical and Numerical Techniques 150
General Guidelines for Exploring Data 153
Data Collection and Sampling 161
5.1
5.2
5.3
5.4
6
ix
Introduction 218
Random Variables and Probability Distributions 218
Bivariate Distributions 229
(Optional) Applications in Finance: Portfolio Diversification
and Asset Allocation 237
Binomial Distribution 243
Poisson Distribution 251
Continuous Probability Distributions 263
8.1
8.2
8.3
8.4
Introduction 264
Probability Density Functions 264
Normal Distribution 270
(Optional) Exponential Distribution 287
Other Continuous Distributions 291
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CONTENTS
9
Sampling Distributions 307
9.1
9.2
9.3
9.4
10
Introduction to Estimation 335
10.1
10.2
10.3
11
Introduction 361
Concepts of Hypothesis Testing 361
Testing the Population Mean When the Population Standard Deviation Is Known 365
Calculating the Probability of a Type II Error 385
The Road Ahead 394
Inference about a Population 398
12.1
12.2
12.3
12.4
13
Introduction 336
Concepts of Estimation 336
Estimating the Population Mean When the Population Standard Deviation Is Known 339
Selecting the Sample Size 353
Introduction to Hypothesis Testing 360
11.1
11.2
11.3
11.4
12
Introduction 308
Sampling Distribution of the Mean 308
Sampling Distribution of a Proportion 321
Sampling Distribution of the Difference between Two Means 327
From Here to Inference 330
Introduction 399
Inference about a Population Mean When the Standard Deviation Is Unknown 399
Inference about a Population Variance 413
Inference about a Population Proportion 421
(Optional) Applications in Marketing: Market Segmentation 435
Inference about Comparing Two Populations 448
13.1
13.2
13.3
13.4
13.5
Appendix 13
Introduction 449
Inference about the Difference between Two Means: Independent Samples 449
Observational and Experimental Data 472
Inference about the Difference between Two Means: Matched Pairs Experiment 475
Inference about the Ratio of Two Variances 489
Inference about the Difference between Two Population Proportions 495
Review of Chapters 12 and 13 519
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CONTENTS
14
Analysis of Variance 525
14.1
14.2
14.3
14.4
14.5
14.6
Appendix 14
15
Introduction 597
Chi-Squared Goodness-of-Fit Test 597
Chi-Squared Test of a Contingency Table 604
Summary of Tests on Nominal Data 615
(Optional) Chi-Squared Test for Normality 617
Review of Chapters 12 to 15 626
Simple Linear Regression and Correlation 633
16.1
16.2
16.3
16.4
16.5
16.6
Appendix 16
17
Introduction 526
One-Way Analysis of Variance 526
Multiple Comparisons 543
Analysis of Variance Experimental Designs 553
Randomized Block (Two-Way) Analysis of Variance 554
Two-Factor Analysis of Variance 563
(Optional) Applications in Operations Management:
Finding and Reducing Variation 578
Review of Chapters 12 to 14 589
Chi-Squared Tests 596
15.1
15.2
15.3
15.4
Appendix 15
16
xi
Introduction 634
Model 634
Estimating the Coefficients 637
Error Variable: Required Conditions 647
Assessing the Model 650
Using the Regression Equation 666
Regression Diagnostics—I 671
Review of Chapters 12 to 16 684
Multiple Regression 692
17.1
17.2
17.3
17.4
Appendix 17
Introduction 693
Model and Required Conditions 693
Estimating the Coefficients and Assessing the Model 694
Regression Diagnostics—II 713
Regression Diagnostics—III (Time Series) 716
Review of Chapters 12 to 17 729
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CONTENTS
Appendix
A
Data File Sample Statistics A-1
Appendix
B
Tables B-1
Appendix
C
Answers to Selected Even-Numbered Exercises C-1
Index I-1
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P r e fa c e
B
usinesses are increasingly using statistical techniques to convert data into information. For students preparing for the business world, it is not enough merely to
focus on mastering a diverse set of statistical techniques and calculations. A course
and its attendant textbook must provide a complete picture of statistical concepts and
their applications to the real world. Statistics for Management and Economics is designed
to demonstrate that statistics methods are vital tools for today’s managers and
economists.
Fulfilling this objective requires the several features that I have built into this book.
First, I have included data-driven examples, exercises, and cases that demonstrate statistical applications that are and can be used by marketing managers, financial analysts,
accountants, economists, operations managers, and others. Many are accompanied by
large and either genuine or realistic data sets. Second, I reinforce the applied nature of
the discipline by teaching students how to choose the correct statistical technique.
Third, I teach students the concepts that are essential to interpreting the statistical
results.
Why I Wrote This Book
Business is complex and requires effective management to succeed. Managing complexity requires many skills. There are more competitors, more places to sell products, and
more places to locate workers. As a consequence, effective decision making is more crucial than ever before. On the other hand, managers have more access to larger and more
detailed data that are potential sources of information. However, to achieve this potential requires that managers know how to convert data into information. This knowledge
extends well beyond the arithmetic of calculating statistics. Unfortunately, this is what
most textbooks offer—a series of unconnected techniques illustrated mostly with manual calculations. This continues a pattern that goes back many years. What is required
is a complete approach to applying statistical techniques.
When I started teaching statistics in 1971, books demonstrated how to calculate
statistics and, in some cases, how various formulas were derived. One reason for doing
so was the belief that by doing calculations by hand, students would be able to understand the techniques and concepts. When the first edition of this book was published in
1988, an important goal was to teach students to identify the correct technique.
Through the next eight editions, I refined my approach to emphasize interpretation
and decision making equally. I now divide the solution of statistical problems into three
stages and include them in every appropriate example: (1) identify the technique,
(2) compute the statistics, and (3) interpret the results. The compute stage can be completed in any or all of three ways: manually (with the aid of a calculator), using Excel
2010, and using Minitab. For those courses that wish to use the computer extensively,
manual calculations can be played down or omitted completely. Conversely, those that
wish to emphasize manual calculations may easily do so, and the computer solutions can
be selectively introduced or skipped entirely. This approach is designed to provide maximum flexibility, and it leaves to the instructor the decision of if and when to introduce
the computer.
xv
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PREFACE
I believe that my approach offers several advantages.
•
An emphasis on identification and interpretation provides students with practical
skills they can apply to real problems they will face regardless of whether a
course uses manual or computer calculations.
•
Students learn that statistics is a method of converting data into information.
With 878 data files and corresponding problems that ask students to interpret
statistical results, students are given ample opportunities to practice data analysis
and decision making.
•
The optional use of the computer allows for larger and more realistic exercises
and examples.
Placing calculations in the context of a larger problem allows instructors to focus on
more important aspects of the decision problem. For example, more attention needs to
be devoted to interpreting statistical results. Proper interpretation of statistical results
requires an understanding of the probability and statistical concepts that underlie the
techniques and an understanding of the context of the problems. An essential aspect of
my approach is teaching students the concepts. I do so in two ways.
1.
Nineteen Java applets allow students to see for themselves how statistical
techniques are derived without going through the sometimes complicated
mathematical derivations.
2.
Instructions are provided about how to create Excel worksheets that allow
students to perform “what-if” analyses. Students can easily see the effect of
changing the components of a statistical technique, such as the effect of
increasing the sample size.
Efforts to teach statistics as a valuable and necessary tool in business and economics are
made more difficult by the positioning of the statistics course in most curricula. The
required statistics course in most undergraduate programs appears in the first or second
year. In many graduate programs, the statistics course is offered in the first semester of
a three-semester program and the first year of a two-year program. Accounting, economics, finance, human resource management, marketing, and operations management
are usually taught after the statistics course. Consequently, most students will not be
able to understand the general context of the statistical application. This deficiency is
addressed in this book by “Applications in . . .” sections, subsections, and boxes.
Illustrations of statistical applications in business that students are unfamiliar with are
preceded by an explanation of the background material.
•
For example, to illustrate graphical techniques, we use an example that compares
the histograms of the returns on two different investments. To explain what
financial analysts look for in the histograms requires an understanding that risk
is measured by the amount of variation in the returns. The example is preceded
by an “Applications in Finance” box that discusses how return on investment is
computed and used.
•
Later when I present the normal distribution, I feature another “Applications in
Finance” box to show why the standard deviation of the returns measures the
risk of that investment.
•
Thirty-six application boxes are scattered throughout the book.
•
I’ve added Do-It-Yourself Excel exercises will teach students to compute
spreadsheets on their own.
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PREFACE
xvii
Some applications are so large that I devote an entire section or subsection to the topic.
For example, in the chapter that introduces the confidence interval estimator of a proportion, I also present market segmentation. In that section, I show how the confidence
interval estimate of a population proportion can yield estimates of the sizes of market
segments. In other chapters, I illustrate various statistical techniques by showing how
marketing managers can apply these techniques to determine the differences that exist
between market segments. There are six such sections and one subsection in this book.
The “Applications in . . .” segments provide great motivation to the student who asks,
“How will I ever use this technique?”
New in This Edition
Six large real data sets are the sources of 150 new exercises. Students will have the
opportunity to convert real data into information. Instructors can use the data sets for
hundreds of additional examples and exercises.
Many of the examples, exercises, and cases using real data in the eighth edition
have been updated. These include the data on wins, payrolls, and attendance in baseball, basketball, football, and hockey; returns on stocks listed on the New York Stock
Exchange, NASDAQ, and Toronto Stock Exchange; and global warming.
Chapter 2 in the eighth edition, which presented graphical techniques, has been
split into two chapters–2 and 3. Chapter 2 describes graphical techniques for nominal
data, and Chapter 3 presents graphical techniques for interval data. Some of the material in the eighth edition Chapter 3 has been incorporated into the new Chapter 3.
To make room for the new additional exercises we have removed Section 12.5,
Applications in Accounting: Auditing.
I’ve created many new examples and exercises. Here are the numbers for the
Abbreviated ninth edition: 116 solved examples, 1727 exercises, 26 cases, 690 data sets,
35 appendixes containing 37 solved examples, 98 exercises, and 25 data sets for a grand
total of 153 solved examples, 1825 exercises, 26 cases, and 715 data sets.
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PREFACE
G U I DE D B O OK TO U R
Data Driven: The Big Picture
Solving statistical problems begins with a problem and
data. The ability to select the right method by problem
objective and data type is a valuable tool for business.
Because business decisions are driven by data, students will
leave this course equipped with the tools they need to make
effective, informed decisions in all areas of the business
world.
EXAMPLE
13.1*
DATA
Xm13-01
Direct and Broker-Purchased Mutual Funds
Millions of investors buy mutual funds (see page 181 for a description of mutual funds),
choosing from thousands of possibilities. Some funds can be purchased directly from
banks or other financial institutions whereas others must be purchased through brokers,
who charge a fee for this service. This raises the question, Can investors do better by
buying mutual funds directly than by purchasing mutual funds through brokers? To
help answer this question, a group of researchers randomly sampled the annual returns
from mutual funds that can be acquired directly and mutual funds that are bought
through brokers and recorded the net annual returns, which are the returns on investment after deducting all relevant fees. These are listed next.
Direct
9.33
6.94
16.17
16.97
5.94
12.61
3.33
16.13
11.2
1.14
Broker
4.68
3.09
7.26
2.05
13.07
0.59
13.57
0.35
2.69
18.45
4.23
10.28
7.1
⫺3.09
5.6
5.27
8.09
15.05
13.21
1.72
14.69
⫺2.97
10.37
⫺0.63
⫺0.15
0.27
4.59
6.38
⫺0.24
10.32
10.29
4.39
⫺2.06
7.66
10.83
14.48
4.8
13.12
⫺6.54
⫺1.06
3.24
⫺6.76
12.8
11.1
2.73
⫺0.13
18.22
⫺0.8
⫺5.75
2.59
3.71
13.15
11.05
⫺3.12
8.94
2.74
4.07
5.6
⫺0.85
⫺0.28
16.4
6.39
⫺1.9
9.49
6.7
0.19
12.39
6.54
10.92
⫺2.15
4.36
⫺11.07
9.24
⫺2.67
8.97
1.87
⫺1.53
5.23
6.87
⫺1.69
9.43
8.31
⫺3.99
⫺4.44
8.63
7.06
1.57
⫺8.44
⫺5.72
6.95
Can we conclude at the 5% significance level that directly purchased mutual funds outperform mutual funds bought through brokers?
SOLUTION
IDENTIFY
To answer the question, we need to compare the population of returns from direct and the
returns from broker-bought mutual funds. The data are obviously interval (we’ve
recorded real numbers). This problem objective–data type combination tells us that the
parameter to be tested is the difference between two means, ␮1 ⫺ ␮2. The hypothesis to
Identify the Correct
Technique
Examples introduce the first
crucial step in this three-step
(identify–compute–interpret)
approach. Every example’s
solution begins by examining
the data type and problem
objective and then identifying
the right technique to solve
the problem.
be tested is that the mean net annual return from directly purchased mutual funds (␮1) is
larger than the mean of broker-purchased funds (␮2). Hence, the alternative hypothesis is
H1: 1m1 - m22 7 0
As usual, the null hypothesis automatically follows:
H0: 1m1 - m22 = 0
To decide which of the t-tests of ␮1 ⫺ ␮2 to apply, we conduct the F-test of s21>s22.
H0: s21>s22 = 1
H1: s21>s22 Z 1
COMPUTE
MANUALLY
From the data, we calculated the following statistics:
s21 = 37.49 and s22 = 43.34
Test statistic: F = s21>s22 = 37.49>43.34 = 0.86
Rejection region: F 7 Fa>2,n ,n = F.025,49,49 L F.025,50,50 = 1.75
1
2
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREFACE
xix
Problem objective?
Describe a population
Data type?
Interval
Nominal
Type of descriptive
measurement?
z-test and
estimator of p
Central location
Variability
t-test and
estimator of m
x 2 -test and
estimator of s 2
APPENDIX 14
REVIEW
OF
Appendixes 13, 14, 15, 16, and 17 reinforce this problem-solving
approach and allow students to hone their skills.
Flowcharts, found within the appendixes, help students develop
the logical process for choosing the correct technique, reinforce
the learning process, and provide easy review material for students.
C H A P T E R S 12
TO
14
The number of techniques introduced in Chapters 12 to 14 is up to 20. As we did in
Appendix 13, we provide a table of the techniques with formulas and required conditions, a flowchart to help you identify the correct technique, and 25 exercises to give
you practice in how to choose the appropriate method. The table and the flowchart
have been amended to include the three analysis of variance techniques introduced in
this chapter and the three multiple comparison methods.
TABLE
A 14 .1
Summary of Statistical Techniques in Chapters 12 to 14
t-test of ␮
Estimator of ␮ (including estimator of N␮)
␹2 test of ␴2
Estimator of ␴2
z-test of p
Estimator of p (including estimator of Np)
Equal-variances t-test of ␮1 ⫺ ␮2
Equal-variances estimator of ␮1 ⫺ ␮2
Unequal-variances t-test of ␮1 ⫺ ␮2
Unequal-variances estimator of ␮1 ⫺ ␮2
t-test of ␮D
Estimator of ␮D
F-test of s21>s22
Estimator of s21>s22
z-test of p1 ⫺ p2 (Case 1)
z-test of p1 ⫺ p2 (Case 2)
Estimator of p1 ⫺ p2
One-way analysis of variance (including multiple comparisons)
Two-way (randomized blocks) analysis of variance
Two-factor analysis of variance
Factors That Identify the t-Test and Estimator of ␮D
1. Problem objective: Compare two populations
2. Data type: Interval
3. Descriptive measurement: Central location
4. Experimental design: Matched pairs
Factors That Identify . . . boxes are found in
each chapter after a technique or concept has
been introduced. These boxes allow students
to see a technique’s essential requirements and
give them a way to easily review their understanding. These essential requirements are
revisited in the review chapters, where they
are coupled with other concepts illustrated in
flowcharts.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREFACE
A GUIDE TO STATISTICAL TECHNIQUES
Problem Objectives
Describe a
Population
Compare Two
Populations
Compare Two or
More Populations
Histogram
Section 3.1
Equal-variances t-test and estimator
of the difference between two
means: independent samples
Section 13.1
One-way analysis of varianc
Section 14.1
Ogive
Section 3.1
Stem-and-leaf
Section 3.1
Interval
Line chart
Section 3.2
Unequal-variances t-test and
estimator of the difference between
two means: independent samples
Section 13.1
Mean, median, and mode
Section 4.1
t-test and estimator of mean
difference
Section 13.3
Range, variance, and
standard deviation
Section
ion 4.2
F-test and estimator of ratio of two
variances
Section 13.4
LSD multiple comparison
method
Section 14.2
Tukey’s multiple compariso
method
Section 14.2
Two-way analysis of varianc
Section 14.4
Two-factor analysis of varia
Section 14.5
A Guide to Statistical Techniques, found
on the inside front cover of the text, pulls
everything together into one useful table
that helps students identify which technique
to perform based on the problem objective
and data type.
Percentiles and quartiles
Section 4.3
Box plot
Section 4.3
DATA T YPES
t-test and estimator of a mean
Section 12.1
Chi-squared test and
estimator of a variance
Section 12.2
Frequency distribution
Section 2.2
Nominal
Bar chart
Section 2.2
Pie chart
Section 2.2
z-test and estimator of the
difference between two proportions
Section 13.5
Chi-squared test of a
contingency table
Section 15.2
Chi-squared test of a contingency
table
Section 15.2
z-test and estimator
of a proportion
Section 12.3
Ordinal
Chi-squared goodness-offit test
Section 15.1
Median
Section 4.1
Percentiles and quartiles
Section 4.3
Box plot
Section 4.3
More Data Sets
A total of 715 data sets available to be downloaded provide ample practice.
These data sets often contain real data, are typically large, and are formatted
for Excel, Minitab, SPSS, SAS, JMP IN, and ASCII.
Prevalent use of data in examples, exercises, and cases is highlighted by
the accompanying data icon, which alerts students to go to Keller’s website.
DATA
Xm13-02
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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xxi
PREFACE
11.38
Xr11-38
Past experience indicates that the monthly
long-distance telephone bill is normally distributed
with a mean of $17.85 and a standard deviation of
$3.87. After an advertising campaign aimed at
increasing long-distance telephone usage, a random
sample of 25 household bills was taken.
a. Do the data allow us to infer at the 10% significance level that the campaign was successful?
b. What assumption must you make to answer
part (a)?
11.39 Xr11-39 In an attempt to reduce the number of per-
son-hours lost as a result of industrial accidents, a
large production plant installed new safety equipment. In a test of the effectiveness of the equipment,
a random sample of 50 departments was chosen.
The number of person-hours lost in the month
before and the month after the installation of the
safety equipment was recorded. The percentage
change was calculated and recorded. Assume that
the population standard deviation is ␴ ⫽ 6. Can we
infer at the 10% significance level that the new
safety equipment is effective?
11.40 Xr11-40 A highway patrol officer believes that the
average speed of cars traveling over a certain stretch
of highway exceeds the posted limit of 55 mph. The
speeds of a random sample of 200 cars were
recorded. Do these data provide sufficient evidence
at the 1% significance level to support the officer’s
belief? What is the p-value of the test? (Assume that
the standard deviation is known to be 5.)
11.41 Xr11-41 An automotive expert claims that the large
number of self-serve gasoline stations has resulted in
poor automobile maintenance, and that the average
tire pressure is more than 4 pounds per square inch
(psi) below its manufacturer’s specification. As a
quick test, 50 tires are examined, and the number of
psi each tire is below specification is recorded. If we
assume that tire pressure is normally distributed
with ␴ ⫽ 1.5 psi, can we infer at the 10% significance level that the expert is correct? What is the
p-value?
11.42 Xr11-42 For the past few years, the number of cus-
tomers of a drive-up bank in New York has averaged
20 per hour, with a standard deviation of 3 per hour.
11.43 Xr11-43 A fast-food franchiser is considering building
a restaurant at a certain location. Based on financial
analyses, a site is acceptable only if the number of
pedestrians passing the location averages
than
E Xmore
AMP
L E 1 3 .9
100 per hour. The number of pedestrians observed
for each of 40 hours was recorded. Assuming that
DATA
the population standard deviation is known to beXm13-09
16,
can we conclude at the 1% significance level that the
site is acceptable?
11.44 Xr11-44 Many Alpine ski centers base their projec-
tions of revenues and profits on the assumption that
the average Alpine skier skis four times per year. To
investigate the validity of this assumption, a random
sample of 63 skiers is drawn and each is asked to
report the number of times he or she skied the previous year. If we assume that the standard deviation
is 2, can we infer at the 10% significance level that
the assumption is wrong?
Test Marketing of Package Designs, Part 1
The General Products Company produces and sells a variety of household products.
Because of stiff competition, one of its products, a bath soap, is not selling well. Hoping
to improve sales, General Products decided to introduce more attractive packaging.
The company’s advertising agency developed two new designs. The first design features
several bright colors to distinguish it from other brands. The second design is light
green in color with just the company’s logo on it. As a test to determine which design is
better, the marketing manager selected two supermarkets. In one supermarket, the soap
was packaged in a box using the first design; in the second supermarket, the second
design was used. The product scanner at each supermarket tracked every buyer of soap
over a 1-week period. The supermarkets recorded the last four digits of the scanner
code for each of the five brands of soap the supermarket sold. The code for the General
Products brand of soap is 9077 (the other codes are 4255, 3745, 7118, and 8855). After
the trial period the scanner data were transferred to a computer file Because the first
11.45 Xr11-45 The golf professional at a private course
claims that members who have taken lessons from
him lowered their handicap by more than five
strokes. The club manager decides to test the claim
by randomly sampling 25 members who have had
lessons and asking each to report the reduction in
handicap, where a negative number indicates an
increase in the handicap. Assuming that the reduction in handicap is approximately normally distributed with a standard deviation of two strokes, test
the golf professional’s claim using a 10% significance level.
11.46 Xr11-46 The current no-smoking regulations in
office buildings require workers who smoke to take
breaks and leave the building in order to satisfy
their habits. A study indicates that such workers
average 32 minutes per day taking smoking breaks.
The standard deviation is 8 minutes. To help reduce
the average break, rooms with powerful exhausts
were installed in the buildings. To see whether these
rooms serve their designed purpose, a random sample of 110 smokers was taken. The total amount of
time away from their desks was measured for 1 day.
Test to determine whether there has been a
decrease in the mean time away from their desks.
Compute the p-value and interpret it relative to the
cute otitis media, an infection of
costs of Type I and Type II errors.
CASE 14.1
Comparing Three Methods of Treating
Childhood Ear Infections*
A
the middle ear, is a common
11.47 Xr11-47 A low-handicap golfer who uses Titleist
childhoodthat
illness.
There are
varibrand golf balls observed
his average
drive
is
ous ways to treat the problem. To help
determine the best way, researchers
conducted an experiment. One hundred
and eighty children between 10 months
and 2 years with recurrent acute otitis
media were divided into three equal
groups. Group 1 was treated by surgically removing the adenoids (adenoidectomy), the second was treated with the
and episodes of acute otitis media were
recorded. The data were recorded in the
following way:
2 years, during which time all symptoms
DATA
C14-01
Column 1: ID number
Column 2: Group number
number of episodes, number of
Column 3: Number of episodes of the
physician visits, number of pre-
illness
scriptions, and number of days
Column 4: Number of visits to a
with symptoms of respiratory
physician because of any infection
Column 5: Number of prescriptions
Column 6: Number of days with
symptoms of respiratory infection
drug Sulfafurazole, and the third with a
placebo. Each child was tracked for
©AP Photo/Chris Carlson
that of 5 years ago, with the possible exception of the
mean, can we conclude at the 5% significance level
that the dean’s claim is true?
a.
infection?
b.
Assume that you are working for
the company that makes the drug
Sulfafurazole. Write a report to the
Are there differences between the
company’s executives discussing
three groups with respect to the
your results.
*This case is adapted from the British Medical Journal, February 2004.
Flexible to Use
Although many texts today incorporate the use of the computer, Statistics for
Management and Economics is designed for maximum flexibility and ease of use for
both instructors and students. To this end, parallel illustration of both manual and computer printouts is provided throughout the text. This approach allows you to choose
which, if any, computer program to use. Regardless of the method or software you
choose, the output and instructions that you need are provided!
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREFACE
Compute the Statistics
Once the correct technique has been identified,
examples take students to the next level within
the solution by asking them to compute the statistics.
COMPUTE
MANUALLY
From the data, we calculated the following statistics:
s21 = 37.49 and s22 = 43.34
Test statistic: F = s21>s22 = 37.49>43.34 = 0.86
Rejection region: F 7 Fa>2,n ,n = F.025,49,49 L F.025,50,50 = 1.75
1
2
or
F 6 F1 - a>2,n ,n = F.975,49,49 = 1>F.025,49,49 L 1>F.025,50,50 = 1>1.75 = .57
1
2
Because F ⫽ .86 is not greater than 1.75 or smaller than .57, we cannot reject the null
hypothesis.
Manual calculation of the problem is presented
first in each “Compute” section of the examples.
EXCEL
1
2
3
4
5
6
7
8
9
10
A
F-Test: Two-Sample for Variances
Mean
Variance
Observations
df
F
P(F<=f) one-tail
F Critical one-tail
B
Direct
6.63
37.49
50
49
0.8650
0.3068
0.6222
C
Broker
3.72
43.34
50
49
The value of the test statistic is F ⫽ .8650. Excel outputs the one-tail p-value. Because
we’re conducting a two-tail test, we double that value. Thus, the p-value of the test we’re
conducting is 2 ⫻ .3068 ⫽ .6136.
Step-by-step instructions in the use of Excel
and Minitab immediately follow the manual
presentation. Instruction appears in the book
with the printouts—there’s no need to incur the
extra expense of separate software manuals.
SPSS and JMP IN are also available at no cost on
the Keller companion website.
INSTRUCTIONS
1. Type or import the data into two columns. (Open Xm13-01.)
2. Click Data, Data Analysis, and F-test Two-Sample for Variances.
3. Specify the Variable 1 Range (A1:A51) and the Variable 2 Range (B1:B51). Type a
value for (.05).
MINITAB
Test for Equal Variances: Direct, Broker
F-Test (Normal Distribution)
Test statistic = 0.86, p-value = 0.614
INSTRUCTIONS
(Note: Some of the printout has been omitted.)
1. Type or import the data into two columns. (Open Xm13-01.)
2. Click Stat, Basic Statistics, and 2 Variances . . . .
3. In the Samples in different columns box, select the First (Direct) and Second
Appendix A
Appendix A provides summary statistics that allow
students to solve applied exercises with data files
by hand. Offering unparalleled flexibility, this feature allows virtually all exercises to be solved by
hand!
D AT A F I L E S A M P L E S T AT I S T I C S
Chapter 10
10.30
10.31
10.32
10.33
10.34
10.35
10.36
10.37
10.38
10.39
10.40
x
x
x
x
x
x
x
x
x
x
x
Chapter 11
=
=
=
=
=
=
=
=
=
=
=
252.38
1,810.16
12.10
10.21
.510
26.81
19.28
15.00
585,063
14.98
27.19
11.35 x = 5,065
11.36 x = 29,120
11 37 x = 569
12.98 n(1) ⫽ 57, n(2) ⫽ 35, n(3) ⫽ 4,
n(4) ⫽ 4
12.100 n(1) ⫽ 245, n(2) ⫽ 745,
n(3) ⫽ 238, n(4) ⫽ 1319, n(5) ⫽ 2453
12.101 n(1) ⫽ 786, n(2) ⫽ 254
12.102 n(1) ⫽ 518, n(2) ⫽ 132
12.124 n(1) ⫽ 81, n(2) ⫽ 47, n(3) ⫽ 167,
n(4) ⫽ 146, n(5) ⫽ 34
12.125 n(1) ⫽ 63, n(2) ⫽ 125, n(3) ⫽ 45,
n(4) ⫽ 87
12.126 n(1) ⫽ 418, n(2) ⫽ 536, n(3) ⫽ 882
12.127 n(1) ⫽ 290, n(2) ⫽ 35
12.128 n(1) ⫽ 72, n(2) ⫽ 77, n(3) ⫽ 37,
n(4) ⫽ 50, n(5) ⫽ 176
12.129 n(1) ⫽ 289, n(2) ⫽ 51
13.28 Planner: x 1 = 6.18, s1 = 1.59,
n1 = 64
Broker: x 2 = 5.94, s2 = 1.61, n2 = 81
13.29 Textbook: x 1 = 63.71, s1 = 5.90,
n1 = 173
No book: x 2 = 66.80, s2 = 6.85,
n2 = 202
13.30 Wendy’s: x 1 = 149.85, s1 = 21.82,
n1 = 213
McDonald’s: x 2 = 154.43, s2 = 23.64,
n2 = 202
13.31 Men: x 1 = 488.4, s1 = 19.6, n1 = 124
Women:x 2 = 498.1,s2 = 21.9,n2 = 187
13.32 Applied: x 1 = 130.93, s1 = 31.99,
n1 = 100
Contacted: x 2 = 126.14, s2 = 26.00,
n = 100
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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xxiii
PREFACE
Flexible Learning
S E E I N G S TAT I S T I C S
For visual learners, the Seeing Statistics feature
refers to online Java applets developed by Gary
McClelland of the University of Colorado, which
use the interactive nature of the web to illustrate
key statistical concepts. With 19 applets and 82
follow-up exercises, students can explore and
interpret statistical concepts, leading them to
greater intuitive understanding. All Seeing
Statistics applets can be found on CourseMate.
applet 17 Plots of Two-Way ANOVA Effects
This applet provides a graph similar to
difference between the row means. The
Applet Exercises
those in Figures 14.5 and 14.6. There are
second slider changes the difference
Label the columns factor A and the rows
three sliders: one for rows, one for
between the column means. The third
factor B. Move the sliders to arrange for
columns, and one for interaction.
slider allows us to see the effects of
each of the following differences. Describe
Moving the top slider changes the
interaction.
what the resulting figure tells you about
differences between levels of factor A,
levels of factor B, and interaction.
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
17.10
ROW
COL
R⫻C
⫺30
0
0
25
30
30
0
0
30
30
0
25
0
⫺30
0
0
20
20
30
30
0
0
⫺20
0
30
⫺30
20
⫺20
30
⫺30
80
Frequency
60
70
Ample use of graphics provides students many opportunities
to see statistics in all its forms. In addition to manually presented figures throughout the text, Excel and Minitab graphic
outputs are given for students to compare to their own results.
60
40
50
20
40
30
0
15
30
45
60
75
Bills
90
105
120
20
10
0
15
30
45
60
75
90
105
120
Long-distance telephone bills
A PPL I E D : B R I D G I N G
THE
GAP
In the real world, it is not enough to know how to generate the statistics. To be truly
effective, a business person must also know how to interpret and articulate the results.
Furthermore, students need a framework to understand and apply statistics within a
realistic setting by using realistic data in exercises, examples, and case studies.
Interpret the Results
INTERPRET
Examples round out the final component of the identify–compute–interpret approach
by asking students to interpret the results in the context of a business-related decision.
This final step motivates and shows how statistics is used in everyday business
situations.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREFACE
An Applied Approach
4 . 5 (O P T I O N A L ) A PPL I C AT I O N S
IN
P RO F E S S I O N A L S P ORT S : B A S E B A L L
In the chapter-opening example, we provided the payrolls and the number of wins from
the 2009 season. We discovered that there is a weak positive linear relationship between
number of wins and payroll. The strength of the linear relationship tells us that some teams
with large payrolls are not successful on the field, whereas some teams with small payrolls
win a large number of games. It would appear that although the amount of money teams
spend is a factor, another factor is how teams spend their money. In this section, we will
analyze the eight seasons between 2002 and 2009 to see how small-payroll teams succeed.
Professional sports in North America is a multibillion-dollar business. The cost of
a new franchise in baseball, football, basketball, and hockey is often in the hundreds of
millions of dollars. Although some teams are financially successful during losing seasons, success on the field is often related to financial success. (Exercises 4.75 and 4.76
Applications in Medicine and Medical Insurance (Optional)
Physicians routinely perform medical tests, called screenings, on their patients.
Screening tests are conducted for all patients in a particular age and gender group,
regardless of their symptoms. For example, men in their 50s are advised to take a
prostate-specific antigen (PSA) test to determine whether there is evidence of prostate
cancer. Women undergo a Pap test for cervical cancer. Unfortunately, few of these tests
are 100% accurate. Most can produce false-positive and false-negative results. A falsepositive result is one in which the patient does not have the disease, but the test shows
positive. A false-negative result is one in which the patient does have the disease, but
the test produces a negative result. The consequences of each test are serious and costly.
A false-negative test results in not detecting a disease in a patient, therefore postponing
treatment, perhaps indefinitely. A false-positive test leads to apprehension and fear for
the patient. In most cases, the patient is required to undergo further testing such as a
biopsy. The unnecessary follow-up procedure can pose medical risks.
False-positive test results have financial repercussions. The cost of the follow-up
procedure, for example, is usually far more expensive than the screening test. Medical
insurance companies as well as government-funded plans are all adversely affected by
false-positive test results. Compounding the problem is that physicians and patients are
incapable of properly interpreting the results. A correct analysis can save both lives and
money.
Bayes’s Law is the vehicle we use to determine the true probabilities associated with
screening tests. Applying the complement rule to the false-positive and false-negative
rates produces the conditional probabilities that represent correct conclusions. Prior
probabilities are usually derived by looking at the overall proportion of people with the
diseases. In some cases, the prior probabilities may themselves have been revised
With Applications in . . . sections and boxes, Statistics
for Management and Economics now includes 45
applications (in finance, marketing, operations
management, human resources, economics, and
accounting) highlighting how statistics is used in those
professions. For example, “Applications in Finance:
Portfolio Diversification and Asset Allocation” shows
how probability is used to help select stocks to minimize risk. A new optional section, “Applications in
Professional Sports: Baseball” contains a subsection on
the success of the Oakland Athletics.
In addition to sections and boxes, Applications in . . .
exercises can be found within the exercise sections to
further reinforce the big picture.
APPLICATIONS in OPERATIONS MANAGEMENT
Quality
A critical aspect of production is quality. The quality of a final product is a
function of the quality of the product’s components. If the components don’t
© Vicki Beaver
fit, the product will not function as planned and likely cease functioning
before its customers expect it to. For example, if a car door is not made to its
specifications, it will not fit. As a result, the door will leak both water and air.
Operations managers attempt to maintain and improve the quality of products by
ensuring that all components are made so that there is as little variation as possible. As
you have already seen, statisticians measure variation by computing the variance.
Incidentally, an entire chapter (Chapter 21) is devoted to the topic of quality.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREFACE
Education and Income: How Are They Related?
DATA
GSS2008*
Chapter-opening examples and solutions
present compelling discussions of how the techniques and concepts introduced in that chapter
are applied to real-world problems. These examples are then revisited with a solution as each
chapter unfolds, applying the methodologies
introduced in the chapter.
If you’re taking this course, you’re probably a student in an undergraduate or
graduate business or economics program. Your plan is to graduate, get a good
© Vicki Beaver
job, and draw a high salary. You have probably assumed that more education
equals better job equals higher income. Is this true? Fortunately, the General Social Survey
recorded two variables that will help determine whether education and income are related and,
xxv
if so, what the value of an additional year of education might be.
On page 663, we will
provide our answer.
Education and Income: How Are They Related?
want to know how education affects income the independent variable is education (EDUC) and
© Vicki Beaver
IDENTIFY
The problem objective is to analyze the relationship between two interval variables. Because we
the dependent variable is income (INCOME).
COMPUTE
EXCEL
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
A
SUMMARY OUTPUT
B
C
D
E
F
Regression Statistics
Multiple R
0.3790
R Square
0.1436
Adjusted R Square
0.1429
Standard Error
35,972
Observations
1189
ANOVA
df
Regression
Residual
Total
SS
1
257,561,051,309
1187 1,535,986,496,000
1188 1,793,547,547,309
Coefficients
–28926
5111
Intercept
EDUC
Standard Error
5117
362
MS
257,561,051,309
1,294,007,158
t Stat
F
199.04
Significance F
6.702E-42
P-value
–5.65 1.971E-08
14.11 6.702E-42
MINITAB
Regression Analysis: INCOME versus EDUC
The regression equation is
Income = –28926 + 5111 EDUC
1189 cases used, 834 cases contain missing values
Predictor
Constant
EDUC
Coef
–28926
5110.7
SE Coef
5117
362.2
T
P
–5.65 0.000
14.11 0.000
S = 35972.3 R-Sq = 14.4% R-Sq(adj) = 14.3%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
1187
1188
SS
2.57561E+11
1.53599E+12
1.79355E+12
MS
2.57561E+11
1294007158
F
199.04
P
0.000
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PREFACE
CASE A15.1
Which Diets Work?
very year, millions of people start
E
each diet. The average weight before
new diets. There is a bewildering
dieting was 220 pounds, and all needed
array of diets to choose from. The
to lose between 30 and 80 pounds. All
question for many people is, which ones
volunteers agreed to follow their diets
work? Researchers at Tufts University in
for 2 months. No exercise or regular
Is there enough evidence to conclude
Boston made an attempt to point
meetings were required. The following
that there are differences between the
dieters in the right direction. Four diets
variables were recorded for each dieter
diets with respect to
were used:
using the format shown here:
1.
2.
Atkins low-carbohydrate diet
Zone high-protein, moderatecarbohydrate diet
3.
Weight Watchers diet
4.
Dr. Ornish’s low-fat diet
Column 6: Quit after 2 months?
1 ⫽ yes, 2 ⫽ no
Column 7: Quit after 1 year? 1 ⫽ yes,
2 ⫽ no
a.
percent weight loss?
Column 1: Identification number
b.
Column 2: Diet
c.
percent HDL increase?
Column 3: Percent weight loss
d.
proportion quitting within
Column 4: Percent low-density lipoprotein
(LDL)—”bad” cholesterol—decrease
DATA
CA15-01
percent LDL decrease?
Many of the examples, exercises,
and cases are based on actual
studies performed by statisticians
and published in journals, newspapers, and magazines, or presented at
conferences. Many data files were
recreated to produce the original
results.
2 months?
e.
proportion quitting after 1 year?
Column 5: Percent high-density lipoproThe study recruited 160 overweight
tein (HDL)—”good” cholesterol—
people and randomly assigned 40 to
increase
A total of 1825 exercises, many of
them new or updated, offer ample
practice for students to use statistics
in an applied context.
CHAPTER SUM M ARY
Histograms are used to describe a single set of interval data.
Statistics practitioners examine several aspects of the shapes
of histograms. These are symmetry, number of modes, and
its resemblance to a bell shape.
We described the difference between time-series data
and cross-sectional data. Time series are graphed by line
charts.
To analyze the relationship between two interval variables, we draw a scatter diagram. We look for the direction
and strength of the linear relationship.
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PREFACE
xxvii
R E S O U RC E S
Learning Resources
The Essential Textbook Resources website: At the Keller website, you’ll find materials previously on the student CD, including: Interactive concept simulation exercises
from Seeing Statistics, the Data Analysis Plus add-in, 715 data sets, optional topics,
and 35 appendixes (for more information, please visit www.cengage.com/bstatistics/
keller).
Student Solutions Manual (ISBN: 1111531889): Students can check their understanding with this manual, which includes worked solutions of even-numbered
exercises from the text.
Seeing Statistics by Gary McClelland: This online product is flexible and addresses
different learning styles. It presents the visual nature of statistical concepts using more
than 150 Java applets to create an intuitive learning environment. Also included are relevant links to examples, exercises, definitions, and search and navigation capabilities.
Teaching Resources
To access both student and faculty resources, please visit www.cengage.com/login.
AC K NOW L E D G M E N T S
Although there is only one name on the cover of this book, the number of people who
made contributions is large. I would like to acknowledge the work of all of them, with
particular emphasis on the following: Paul Baum, California State University,
Northridge, and John Lawrence, California State University, Fullerton, reviewed the
page proofs. Their job was to find errors in presentation, arithmetic, and composition.The following individuals played important roles in the production of this book
Senior Acquisitions Editor: Charles McCormick, Jr., Developmental Editor: Elizabeth
Lowry, Content Project Manager: Jacquelyn K Featherly and Project Manager Lynn
Lustberg (For all remaining errors, place the blame where it belongs–on me.) Their
advice and suggestions made my task considerably easier.
Fernando Rodriguez produced the test bank stored on the Instructor’s Suite
CD-ROM.
Trent Tucker, Wilfrid Laurier University, and Zvi Goldstein, California State
University, Fullerton, each produced a set of PowerPoint slides.
The author extends thanks also to the survey participants and reviewers of the
previous editions: Paul Baum, California State University, Northridge; Nagraj
Balakrishnan, Clemson University; Howard Clayton, Auburn University; Philip
Cross, Georgetown University; Barry Cuffe, Wingate University; Ernest Demba,
Washington University–St. Louis; Neal Duffy, State University of New York,
Plattsburgh; John Dutton, North Carolina State University; Erick Elder, University
of Arkansas; Mohammed El-Saidi, Ferris State University; Grace Esimai, University
of Texas at Arlington; Abe Feinberg, California State University, Northridge; Samuel
Graves, Boston College; Robert Gould, UCLA; John Hebert, Virginia Tech; James
Hightower, California State University, Fullerton; Bo Honore, Princeton University;
Onisforos Iordanou, Hunter College; Gordon Johnson, California State University,
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PREFACE
Northridge; Hilke Kayser, Hamilton College; Kenneth Klassen, California State
University, Northridge; Roger Kleckner, Bowling Green State University–Firelands;
Harry Kypraios, Rollins College; John Lawrence, California State University,
Fullerton; Dennis Lin, Pennsylvania State University; Neal Long, Stetson
University; George Marcoulides, California State University, Fullerton; Paul Mason,
University of North Florida; Walter Mayer, University of Mississippi; John
McDonald, Flinders University; Richard McGowan, Boston College; Richard
McGrath, Bowling Green State University; Amy Miko, St. Francis College; Janis
Miller, Clemson University; Glenn Milligan, Ohio State University; James Moran,
Oregon State University; Patricia Mullins, University of Wisconsin; Kevin Murphy,
Oakland University; Pin Ng, University of Illinois; Des Nicholls, Australian National
University; Andrew Paizis, Queens College; David Pentico, Duquesne University; Ira
Perelle, Mercy College; Nelson Perera, University of Wollongong; Amy Puelz,
Southern Methodist University; Lawrence Ries, University of Missouri; Colleen
Quinn, Seneca College; Tony Quon, University of Ottawa; Madhu Rao, Bowling
Green State University; Phil Roth, Clemson University; Farhad Saboori, Albright
College; Don St. Jean, George Brown College; Hedayeh Samavati, Indiana–Purdue
University; Sandy Shroeder, Ohio Northern University; Jineshwar Singh, George
Brown College; Natalia Smirnova, Queens College; Eric Sowey, University of New
South Wales; Cyrus Stanier, Virginia Tech; Stan Stephenson, Southwest Texas State
University; Arnold Stromberg, University of Kentucky; Steve Thorpe, University of
Northern Iowa; Sheldon Vernon, Houston Baptist University; and W. F. Younkin,
University of Miami.
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Statistics
FOR MANAGEMENT AND ECONOMICS ABBREVIATED
9e
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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© sellingpix/Shutterstock
WHAT IS STATISTICS?
1.1
Key Statistical Concepts
1.2
Statistical Applications in Business
1.3
Large Real Data Sets
1.4
Statistics and the Computer
Appendix 1
I N T RO D U C T I O N
Instructions for Keller’s website
S
tatistics is a way to get information from data. That’s it! Most of this textbook is
devoted to describing how, when, and why managers and statistics practitioners*
conduct statistical procedures. You may ask, “If that’s all there is to statistics, why is
this book (and most other statistics books) so large?” The answer is that students of
applied statistics will be exposed to different kinds of information and data. We demonstrate some of these with a case and two examples that are featured later in this book.
The first may be of particular interest to you.
*The term statistician is used to describe so many different kinds of occupations that it has ceased to have
any meaning. It is used, for example, to describe a person who calculates baseball statistics as well as an
individual educated in statistical principles. We will describe the former as a statistics practitioner and the
(continued )
1
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 1
EXAMPLE
3.3
Business Statistics Marks (See Chapter 3)
A student enrolled in a business program is attending his first class of the required statistics course. The student is somewhat apprehensive because he believes the myth that the
course is difficult. To alleviate his anxiety, the student asks the professor about last year’s
marks. Because this professor is friendly and helpful, like all other statistics professors, he
obliges the student and provides a list of the final marks, which are composed of term
work plus the final exam. What information can the student obtain from the list?
This is a typical statistics problem. The student has the data (marks) and needs to
apply statistical techniques to get the information he requires. This is a function of
descriptive statistics.
Descriptive Statistics
Descriptive statistics deals with methods of organizing, summarizing, and presenting
data in a convenient and informative way. One form of descriptive statistics uses graphical techniques that allow statistics practitioners to present data in ways that make it
easy for the reader to extract useful information. In Chapters 2 and 3 we will present a
variety of graphical methods.
Another form of descriptive statistics uses numerical techniques to summarize data.
One such method that you have already used frequently calculates the average or mean.
In the same way that you calculate the average age of the employees of a company, we
can compute the mean mark of last year’s statistics course. Chapter 4 introduces several
numerical statistical measures that describe different features of the data.
The actual technique we use depends on what specific information we would like to
extract. In this example, we can see at least three important pieces of information. The first
is the “typical” mark. We call this a measure of central location. The average is one such measure. In Chapter 4, we will introduce another useful measure of central location, the
median. Suppose the student was told that the average mark last year was 67. Is this enough
information to reduce his anxiety? The student would likely respond “No” because he
would like to know whether most of the marks were close to 67 or were scattered far below
and above the average. He needs a measure of variability. The simplest such measure is the
range, which is calculated by subtracting the smallest number from the largest. Suppose the
largest mark is 96 and the smallest is 24. Unfortunately, this provides little information
since it is based on only two marks. We need other measures—these will be introduced in
Chapter 4. Moreover, the student must determine more about the marks. In particular, he
needs to know how the marks are distributed between 24 and 96. The best way to do this is
to use a graphical technique, the histogram, which will be introduced in Chapter 3.
latter as a statistician. A statistics practitioner is a person who uses statistical techniques properly.
Examples of statistics practitioners include the following:
1. a financial analyst who develops stock portfolios based on historical rates of return;
2. an economist who uses statistical models to help explain and predict variables such as inflation rate,
unemployment rate, and changes in the gross domestic product; and
3. a market researcher who surveys consumers and converts the responses into useful information.
Our goal in this book is to convert you into one such capable individual.
The term statistician refers to an individual who works with the mathematics of statistics. His or
her work involves research that develops techniques and concepts that in the future may help the statistics practitioner. Statisticians are also statistics practitioners, frequently conducting empirical research
and consulting. If you’re taking a statistics course, your instructor is probably a statistician.
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WHAT IS STATISTICS?
3
Case 12.1 Pepsi’s Exclusivity Agreement with a University (see Chapter 12)
In the last few years, colleges and universities have signed exclusivity agreements with a
variety of private companies. These agreements bind the university to sell these companies’ products exclusively on the campus. Many of the agreements involve food and
beverage firms.
A large university with a total enrollment of about 50,000 students has offered
Pepsi-Cola an exclusivity agreement that would give Pepsi exclusive rights to sell its
products at all university facilities for the next year with an option for future years. In
return, the university would receive 35% of the on-campus revenues and an additional
lump sum of $200,000 per year. Pepsi has been given 2 weeks to respond.
The management at Pepsi quickly reviews what it knows. The market for soft
drinks is measured in terms of 12-ounce cans. Pepsi currently sells an average of
22,000 cans per week over the 40 weeks of the year that the university operates. The
cans sell for an average of one dollar each. The costs, including labor, total 30 cents per
can. Pepsi is unsure of its market share but suspects it is considerably less than 50%. A
quick analysis reveals that if its current market share were 25%, then, with an exclusivity agreement, Pepsi would sell 88,000 (22,000 is 25% of 88,000) cans per week or
3,520,000 cans per year. The gross revenue would be computed as follows†:
Gross revenue ⫽ 3,520,000 ⫻ $1.00/can ⫽ $3,520,000
This figure must be multiplied by 65% because the university would rake in 35%
of the gross. Thus,
Gross revenue after deducting 35% university take
⫽ 65% ⫻ $3,520,000 ⫽ $2,288,000
The total cost of 30 cents per can (or $1,056,000) and the annual payment to the
university of $200,000 are subtracted to obtain the net profit:
Net profit ⫽ $2,288,000 ⫺ $1,056,000 ⫺ $200,000 ⫽ $1,032,000
Pepsi’s current annual profit is
40 weeks ⫻ 22,000 cans/week ⫻ $.70 ⫽ $616,000
If the current market share is 25%, the potential gain from the agreement is
$1,032,000 – $616,000 ⫽ $416,000
The only problem with this analysis is that Pepsi does not know how many soft drinks
are sold weekly at the university. Coke is not likely to supply Pepsi with information about
its sales, which together with Pepsi’s line of products constitute virtually the entire market.
Pepsi assigned a recent university graduate to survey the university’s students to supply the missing information. Accordingly, she organizes a survey that asks 500 students to
keep track of the number of soft drinks they purchase in the next 7 days. The responses
are stored in a file C12-01 available to be downloaded. See Appendix 1 for instructions.
Inferential Statistics
The information we would like to acquire in Case 12.1 is an estimate of annual profits
from the exclusivity agreement. The data are the numbers of cans of soft drinks consumed in 7 days by the 500 students in the sample. We can use descriptive techniques to
†We have created an Excel spreadsheet that does the calculations for this case. See Appendix 1 for
instructions on how to download this spreadsheet from Keller’s website plus hundreds of datasets and
much more.
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CHAPTER 1
learn more about the data. In this case, however, we are not so much interested in what
the 500 students are reporting as in knowing the mean number of soft drinks consumed
by all 50,000 students on campus. To accomplish this goal we need another branch of
statistics: inferential statistics.
Inferential statistics is a body of methods used to draw conclusions or inferences
about characteristics of populations based on sample data. The population in question
in this case is the university’s 50,000 students. The characteristic of interest is the soft
drink consumption of this population. The cost of interviewing each student in the
population would be prohibitive and extremely time consuming. Statistical techniques
make such endeavors unnecessary. Instead, we can sample a much smaller number of
students (the sample size is 500) and infer from the data the number of soft drinks consumed by all 50,000 students. We can then estimate annual profits for Pepsi.
Example
12.5
Exit Polls (see Chapter 12)
When an election for political office takes place, the television networks cancel regular
programming to provide election coverage. After the ballots are counted, the results are
reported. However, for important offices such as president or senator in large states, the
networks actively compete to see which one will be the first to predict a winner. This is
done through exit polls in which a random sample of voters who exit the polling booth
are asked for whom they voted. From the data, the sample proportion of voters supporting the candidates is computed. A statistical technique is applied to determine whether
there is enough evidence to infer that the leading candidate will garner enough votes to
win. Suppose that the exit poll results from the state of Florida during the year 2000 elections were recorded. Although several candidates were running for president, the exit
pollsters recorded only the votes of the two candidates who had any chance of winning:
Republican George W. Bush and Democrat Albert Gore. The results (765 people who
voted for either Bush or Gore) were stored in file Xm12-05. The network analysts would
like to know whether they can conclude that George W. Bush will win the state of Florida.
Example 12.5 describes a common application of statistical inference. The population the television networks wanted to make inferences about is the approximately
5 million Floridians who voted for Bush or Gore for president. The sample consisted of
the 765 people randomly selected by the polling company who voted for either of the
two main candidates. The characteristic of the population that we would like to know is
the proportion of the Florida total electorate that voted for Bush. Specifically, we would
like to know whether more than 50% of the electorate voted for Bush (counting only
those who voted for either the Republican or Democratic candidate). It must be made
clear that we cannot predict the outcome with 100% certainty because we will not ask
all 5 million actual voters for whom they voted. This is a fact that statistics practitioners
and even students of statistics must understand. A sample that is only a small fraction of
the size of the population can lead to correct inferences only a certain percentage of the
time. You will find that statistics practitioners can control that fraction and usually set it
between 90% and 99%.
Incidentally, on the night of the United States election in November 2000, the networks goofed badly. Using exit polls as well as the results of previous elections, all four
networks concluded at about 8 P.M. that Al Gore would win Florida. Shortly after
10 P.M., with a large percentage of the actual vote having been counted, the networks
reversed course and declared that George W. Bush would win the state. By 2 A.M.,
another verdict was declared: The result was too close to call. In the future, this experience will likely be used by statistics instructors when teaching how not to use statistics.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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WHAT IS STATISTICS?
5
Notice that, contrary to what you probably believed, data are not necessarily numbers. The marks in Example 3.3 and the number of soft drinks consumed in a week in
Case 12.1, of course, are numbers; however, the votes in Example 12.5 are not. In
Chapter 2, we will discuss the different types of data you will encounter in statistical
applications and how to deal with them.
1.1
K E Y S TAT I S T I C A L C O N C E P T S
Statistical inference problems involve three key concepts: the population, the sample,
and the statistical inference. We now discuss each of these concepts in more detail.
Population
A population is the group of all items of interest to a statistics practitioner. It is frequently very large and may, in fact, be infinitely large. In the language of statistics, population does not necessarily refer to a group of people. It may, for example, refer to the
population of ball bearings produced at a large plant. In Case 12.1, the population of
interest consists of the 50,000 students on campus. In Example 12.5, the population
consists of the Floridians who voted for Bush or Gore.
A descriptive measure of a population is called a parameter. The parameter of
interest in Case 12.1 is the mean number of soft drinks consumed by all the students at
the university. The parameter in Example 12.5 is the proportion of the 5 million
Florida voters who voted for Bush. In most applications of inferential statistics the parameter represents the information we need.
Sample
A sample is a set of data drawn from the studied population. A descriptive measure of a
sample is called a statistic. We use statistics to make inferences about parameters. In
Case 12.1, the statistic we would compute is the mean number of soft drinks consumed in
the last week by the 500 students in the sample. We would then use the sample mean to
infer the value of the population mean, which is the parameter of interest in this problem.
In Example 12.5, we compute the proportion of the sample of 765 Floridians who voted
for Bush. The sample statistic is then used to make inferences about the population of all
5 million votes—that is, we predict the election results even before the actual count.
Statistical Inference
Statistical inference is the process of making an estimate, prediction, or decision about
a population based on sample data. Because populations are almost always very large,
investigating each member of the population would be impractical and expensive. It is far
easier and cheaper to take a sample from the population of interest and draw conclusions
or make estimates about the population on the basis of information provided by the sample. However, such conclusions and estimates are not always going to be correct. For this
reason, we build into the statistical inference a measure of reliability. There are two such
measures: the confidence level and the significance level. The confidence level is the
proportion of times that an estimating procedure will be correct. For example, in
Case 12.1, we will produce an estimate of the average number of soft drinks to be
consumed by all 50,000 students that has a confidence level of 95%. In other words,
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estimates based on this form of statistical inference will be correct 95% of the time.
When the purpose of the statistical inference is to draw a conclusion about a population,
the significance level measures how frequently the conclusion will be wrong. For example,
suppose that, as a result of the analysis in Example 12.5, we conclude that more than
50% of the electorate will vote for George W. Bush, and thus he will win the state of
Florida. A 5% significance level means that samples that lead us to conclude that Bush
wins the election, will be wrong 5% of the time.
1. 2 S TAT I S T I C A L A PPL I C AT I O N S
IN
BUSINESS
An important function of statistics courses in business and economics programs is to
demonstrate that statistical analysis plays an important role in virtually all aspects of
business and economics. We intend to do so through examples, exercises, and cases.
However, we assume that most students taking their first statistics course have not
taken courses in most of the other subjects in management programs. To understand
fully how statistics is used in these and other subjects, it is necessary to know something
about them. To provide sufficient background to understand the statistical application
we introduce applications in accounting, economics, finance, human resources management, marketing, and operations management. We will provide readers with some
background to these applications by describing their functions in two ways.
Application Sections and Subsections
We feature five sections that describe statistical applications in the functional areas of
business. For example, in Section 7.3 we show an application in finance that describes a
financial analyst’s use of probability and statistics to construct portfolios that decrease risk.
One section and one subsection demonstrate the uses of probability and statistics in
specific industries. Section 4.5 introduces an interesting application of statistics in professional baseball. A subsection in Section 6.4 presents an application in medical testing
(useful in the medical insurance industry).
Application Boxes
For other topics that require less detailed description, we provide application boxes
with a relatively brief description of the background followed by examples or exercises.
These boxes are scattered throughout the book. For example, in Chapter 3 we discuss a
job a marketing manager may need to undertake to determine the appropriate price for
a product. To understand the context, we need to provide a description of marketing
management. The statistical application will follow.
1.3
L A RG E R E A L D ATA S E T S
The authors believe that you learn statistics by doing statistics. For their lives after college and university, we expect our students to have access to large amounts of real data
that must be summarized to acquire the information needed to make decisions. To provide practice in this vital skill we have created six large real datasets, available to be
downloaded from Keller’s website. Their sources are the General Social Survey (GSS)
and the American National Election Survey (ANES).
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7
General Social Survey
Since 1972, the General Social Survey has been tracking American attitudes on a wide
variety of topics. Except for the United States census, the GSS is the most frequently
used sources of information about American society. The surveys now conducted every
second year measure hundreds of variables and thousands of observations. We have
included the results of the last four surveys (years 2002, 2004, 2006, and 2008) stored as
GSS2002, GSS2004, GSS2006, and GSS2008, respectively. The survey sizes are 2,765,
2,812, 4,510, and 2,023, respectively. We have reduced the number of variables to about
60 and have deleted the responses that are known as missing data (“Don’t know,”
“Refused,” etc.).
We have included some demographic variables such as, age, gender, race, income,
and education. Others measure political views, support for various government activities, and work. The full lists of variables for each year are stored in Appendixes
GSS2002, GSS2004, GSS2006, and GSSS2008 that can be downloaded from Keller’s
website.
We have scattered throughout this book examples and exercises drawn from these
data sets.
American National Election Survey
The goal of the American National Election Survey is to provide data about why
Americans vote as they do. The surveys are conducted in presidential election years. We
have included data from the 2004 and 2008 surveys. Like the General Social Survey, the
ANES includes demographic variables. It also deals with interest in the presidential election as well as variables describing political beliefs and affiliations. Online Appendixes
ANES2004 and ANES2008 contain the names and definitions of the variables.
The 2008 surveys overly sampled African American and Hispanic voters. We have
“adjusted” these data by randomly deleting responses from these two racial groups.
As is the case with the General Social Surveys, we have removed missing data.
1.4 S TAT I S T I C S
AND THE
CO M PU T E R
In virtually all applications of statistics, the statistics practitioner must deal with large
amounts of data. For example, Case 12.1 (Pepsi-Cola) involves 500 observations. To
estimate annual profits, the statistics practitioner would have to perform computations
on the data. Although the calculations do not require any great mathematical skill, the
sheer amount of arithmetic makes this aspect of the statistical method time-consuming
and tedious.
Fortunately, numerous commercially prepared computer programs are available to
perform the arithmetic. We have chosen to use Microsoft Excel, which is a spreadsheet
program, and Minitab, which is a statistical software package. (We use the latest versions of both software: Office 2010 and Minitab 16.) We chose Excel because we believe
that it is and will continue to be the most popular spreadsheet package. One of its drawbacks is that it does not offer a complete set of the statistical techniques we introduce in
this book. Consequently, we created add-ins that can be loaded onto your computer to
enable you to use Excel for all statistical procedures introduced in this book. The addins can be downloaded and, when installed, will appear as Data Analysis Plus© on Excel’s
Add-Ins menu. Also available are introductions to Excel and Minitab, and detailed
instructions for both software packages.
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Appendix 1 describes the material that can be downloaded and provides instructions on how to acquire the various components.
A large proportion of the examples, exercises, and cases feature large data sets.
These are denoted with the file name on an orange background. We demonstrate the
solution to the statistical examples in three ways: manually, by employing Excel, and by
using Minitab. Moreover, we will provide detailed instructions for all techniques.
The files contain the data needed to produce the solution. However, in many real
applications of statistics, additional data are collected. For instance, in Example 12.5,
the pollster often records the voter’s gender and asks for other information including
race, religion, education, and income. Many other data sets are similarly constructed. In
later chapters, we will return to these files and require other statistical techniques to
extract the needed information. (Files that contain additional data are denoted by an
asterisk on the file name.)
The approach we prefer to take is to minimize the time spent on manual computations
and to focus instead on selecting the appropriate method for dealing with a problem and on
interpreting the output after the computer has performed the necessary computations. In
this way, we hope to demonstrate that statistics can be as interesting and as practical as any
other subject in your curriculum.
Applets and Spreadsheets
Books written for statistics courses taken by mathematics or statistics majors are
considerably different from this one. It is not surprising that such courses feature mathematical proofs of theorems and derivations of most procedures. When the material is
covered in this way, the underlying concepts that support statistical inference are
exposed and relatively easy to see. However, this book was created for an applied course
in business and economics statistics. Consequently, we do not address directly the
mathematical principles of statistics. However, as we pointed out previously, one of the
most important functions of statistics practitioners is to properly interpret statistical
results, whether produced manually or by computer. And, to correctly interpret statistics,
students require an understanding of the principles of statistics.
To help students understand the basic foundation, we offer two approaches. First,
we will teach readers how to create Excel spreadsheets that allow for what-if analyses.
By changing some of the input value, students can see for themselves how statistics
works. (The term is derived from what happens to the statistics if I change this value?)
These spreadsheets can also be used to calculate many of the same statistics that we
introduce later in this book. Second, we offer applets, which are computer programs that
perform similar what-if analyses or simulations. The applets and the spreadsheet applications appear in several chapters and explained in greater detail.
CHAPTER SUM M ARY
IMPORTANT TERMS
Descriptive statistics 2
Inferential statistics 4
Exit polls 4
Population 5
Parameter 5
Sample 5
Statistic 5
Statistical inference 5
Confidence level 5
Significance level 5
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9
CHAPTER EXERCISES
1.1 In your own words, define and give an example of
each of the following statistical terms.
a.
b.
c.
d.
e.
population
sample
parameter
statistic
statistical inference
l.2 Briefly describe the difference between descriptive
heads and tails when flipped a very large number of
times.
a.
b.
c.
d.
e.
f.
statistics and inferential statistics.
1.3 A politician who is running for the office of mayor of
a city with 25,000 registered voters commissions a
survey. In the survey, 48% of the 200 registered voters interviewed say they plan to vote for her.
a. What is the population of interest?
b. What is the sample?
c. Is the value 48% a parameter or a statistic?
Explain.
1.4 A manufacturer of computer chips claims that less
than 10% of its products are defective. When 1,000
chips were drawn from a large production, 7.5%
were found to be defective.
a.
b.
c.
d.
e.
What is the population of interest?
What is the sample?
What is the parameter?
What is the statistic?
Does the value 10% refer to the parameter or to
the statistic?
f. Is the value 7.5% a parameter or a statistic?
g. Explain briefly how the statistic can be used to
make inferences about the parameter to test the
claim.
1.5 Suppose you believe that, in general, graduates who
have majored in your subject are offered higher
salaries upon graduating than are graduates of other
programs. Describe a statistical experiment that
could help test your belief.
1.6 You are shown a coin that its owner says is fair in
the sense that it will produce the same number of
Describe an experiment to test this claim.
What is the population in your experiment?
What is the sample?
What is the parameter?
What is the statistic?
Describe briefly how statistical inference can be
used to test the claim.
1.7 Suppose that in Exercise 1.6 you decide to flip the
coin 100 times.
a. What conclusion would you be likely to draw if
you observed 95 heads?
b. What conclusion would you be likely to draw if
you observed 55 heads?
c. Do you believe that, if you flip a perfectly fair coin
100 times, you will always observe exactly 50 heads?
If you answered “no,” then what numbers do you
think are possible? If you answered “yes,” how
many heads would you observe if you flipped the
coin twice? Try flipping a coin twice and repeating
this experiment 10 times and report the results.
1.8 Xm01-08 The owner of a large fleet of taxis is trying
to estimate his costs for next year’s operations. One
major cost is fuel purchases. To estimate fuel purchases, the owner needs to know the total distance
his taxis will travel next year, the cost of a gallon of
fuel, and the fuel mileage of his taxis. The owner
has been provided with the first two figures (distance estimate and cost of a gallon of fuel).
However, because of the high cost of gasoline, the
owner has recently converted his taxis to operate
on propane. He has measured and recorded the
propane mileage (in miles per gallon) for 50 taxis.
a.
b.
c.
d.
e.
What is the population of interest?
What is the parameter the owner needs?
What is the sample?
What is the statistic?
Describe briefly how the statistic will produce
the kind of information the owner wants.
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APPENDIX 1
I N S T RU C T I O N S
F OR
K E L L E R’ S
WEBSITE
The Keller website that accompanies this book contains the following features:
Data Analysis Plus 9.0 in VBA, which works with new and earlier versions of Excel
(Office 1997, 2000, XP, 2003, 2007, and 2010 Office for Mac 2004)
A help file for Data Analysis Plus 9.0 in VBA
Data files in the following formats: ASCII, Excel, JMP, Minitab, SAS, and SPSS
Excel workbooks
Seeing Statistics (Java applets that teach a number of important statistical concepts)
Appendices (40 additional topics that are not covered in the book)
Formula card listing every formula in the book
Keller website Instructions
“Data Analysis Plus 9.0 in VBA” can be found on the Keller website. It will be installed
into the XLSTART folder of the most recent version of Excel on your computer. If
properly installed Data Analysis Plus will be a menu item in Excel. The help file for
Data Analysis Plus will be stored directly in your computer’s My Documents folder. It
will appear when you click the Help button or when you make a mistake when using
Data Analysis Plus.
The Data Sets will also be installed from a link within the Keller website.
The Excel workbooks, Seeing Statistics Applets, and Appendixes will be accessed
from the Keller website. Alternatively, you can store the Excel workbooks and
Appendixes to your hard drive.
The Keller website is available using the student access code accompanying all
new books. For more information on how to access the Keller website, please visit
www.cengage.com/bstatistics/keller.
For technical support, please visit www.cengage.com/support for contact options.
Refer to Statistics for Management and Economics, Ninth edition, by Gerald Keller
(ISBN 0-538-47749-0).
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2
© Steve Cole/Digital Vision/Getty Images
Graphical Descriptive
Techniques I
Types of Data and Information
2.2
Describing a Set of Nominal Data
2.3
Describing the Relationship between Two Nominal Variables and Comparing
Two or More Nominal Data Sets
Do Male and Female American Voters Differ in
Their Party Affiliation?
DATA
ANES2008*
In Chapter 1, we introduced the American National Election Survey (ANES), which is
conducted every 4 years with the objective of developing information about how
Americans vote. One question in the 2008 survey was “Do you think of yourself as
Democrat, Republican, Independent, or what?”
Responses were
1. Democrat
© AP Photo/David Smith
2.1
On page 37 we will
provide our answer.
2. Republican
3. Independent
11
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4. Other party
5. No preference
Respondents were also identified by gender: 1 male, and 2 female. The responses are stored in file ANES2008* on our
Keller’s website. The asterisk indicates that there are variables that are not needed for this example but which will be used
later in this book. For Excel users, GENDER AND PARTY are in columns B and BD, respectively. For Minitab users, GENDER AND
PARTY are in columns 2 and 56, respectively. Some of the data are listed here.
ID
GENDER
PARTY
1
1
3
2
2
1
3
2
2
.
.
.
.
.
.
1795
1
1
1796
1
2
1797
1
1
Determine whether American female and male voters differ in their political affiliations.
I N T RO D U C T I O N
I
n Chapter 1, we pointed out that statistics is divided into two basic areas: descriptive
statistics and inferential statistics. The purpose of this chapter, together with the
next, is to present the principal methods that fall under the heading of descriptive
statistics. In this chapter, we introduce graphical and tabular statistical methods that
allow managers to summarize data visually to produce useful information that is often
used in decision making. Another class of descriptive techniques, numerical methods, is
introduced in Chapter 4.
Managers frequently have access to large masses of potentially useful data. But
before the data can be used to support a decision, they must be organized and summarized. Consider, for example, the problems faced by managers who have access to the
databases created by the use of debit cards. The database consists of the personal information supplied by the customer when he or she applied for the debit card. This information includes age, gender, residence, and the cardholder’s income. In addition, each
time the card is used the database grows to include a history of the timing, price, and
brand of each product purchased. Using the appropriate statistical technique, managers
can determine which segments of the market are buying their company’s brands.
Specialized marketing campaigns, including telemarketing, can be developed. Both
descriptive and inferential statistics would likely be employed in the analysis.
Descriptive statistics involves arranging, summarizing, and presenting a set of
data in such a way that useful information is produced. Its methods make use of graphical techniques and numerical descriptive measures (such as averages) to summarize and
present the data, allowing managers to make decisions based on the information generated. Although descriptive statistical methods are quite straightforward, their importance should not be underestimated. Most management, business, and economics
students will encounter numerous opportunities to make valuable use of graphical and
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13
numerical descriptive techniques when preparing reports and presentations in the
workplace. According to a Wharton Business School study, top managers reach a consensus 25% more quickly when responding to a presentation in which graphics are
used.
In Chapter 1, we introduced the distinction between a population and a sample.
Recall that a population is the entire set of observations under study, whereas a sample
is a subset of a population. The descriptive methods presented in this chapter and in
Chapters 3 and 4 apply to both a set of data constituting a population and a set of data
constituting a sample.
In both the preface and Chapter 1, we pointed out that a critical part of your education as statistics practitioners includes an understanding of not only how to draw
graphs and calculate statistics (manually or by computer) but also when to use each technique that we cover. The two most important factors that determine the appropriate
method to use are (1) the type of data and (2) the information that is needed. Both are
discussed next.
2 . 1 T Y PE S
OF
D ATA
AND
I N F OR M AT I O N
The objective of statistics is to extract information from data. There are different types
of data and information. To help explain this important principle, we need to define
some terms.
A variable is some characteristic of a population or sample. For example, the mark
on a statistics exam is a characteristic of statistics exams that is certainly of interest to
readers of this book. Not all students achieve the same mark. The marks will vary from
student to student, thus the name variable. The price of a stock is another variable. The
prices of most stocks vary daily. We usually represent the name of the variable using
uppercase letters such as X, Y, and Z.
The values of the variable are the possible observations of the variable. The values
of statistics exam marks are the integers between 0 and 100 (assuming the exam is
marked out of 100). The values of a stock price are real numbers that are usually measured in dollars and cents (sometimes in fractions of a cent). The values range from 0 to
hundreds of dollars.
Data* are the observed values of a variable. For example, suppose that we observe
the following midterm test marks of 10 students:
67
74
71
83
93
55
48
82
68
62
These are the data from which we will extract the information we seek. Incidentally,
data is plural for datum. The mark of one student is a datum.
When most people think of data, they think of sets of numbers. However, there are
three types of data: interval, nominal, and ordinal.†
*Unfortunately, the term data, like the term statistician, has taken on several different meanings. For
example, dictionaries define data as facts, information, or statistics. In the language of computers, data
may refer to any piece of information such as this textbook or an essay you have written. Such definitions make it difficult for us to present statistics as a method of converting data into information. In this
book, we carefully distinguish among the three terms.
†There
are actually four types of data, the fourth being ratio data. However, for statistical purposes there
is no difference between ratio and interval data. Consequently, we combine the two types.
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Interval data are real numbers, such as heights, weights, incomes, and distances.
We also refer to this type of data as quantitative or numerical.
The values of nominal data are categories. For example, responses to questions
about marital status produce nominal data. The values of this variable are single, married, divorced, and widowed. Notice that the values are not numbers but instead are
words that describe the categories. We often record nominal data by arbitrarily assigning a number to each category. For example, we could record marital status using the
following codes:
single 1, married 2, divorced 3, widowed 4
However, any other numbering system is valid provided that each category has a different number assigned to it. Here is another coding system that is just as valid as the
previous one.
Single 7, married 4, divorced 13, widowed 1
Nominal data are also called qualitative or categorical.
The third type of data is ordinal. Ordinal data appear to be nominal, but the difference is that the order of their values has meaning. For example, at the completion of
most college and university courses, students are asked to evaluate the course. The variables are the ratings of various aspects of the course, including the professor. Suppose
that in a particular college the values are
poor, fair, good, very good, and excellent
The difference between nominal and ordinal types of data is that the order of the values
of the latter indicate a higher rating. Consequently, when assigning codes to the values,
we should maintain the order of the values. For example, we can record the students’
evaluations as
Poor 1, Fair 2, Good 3, Very good 4, Excellent 5
Because the only constraint that we impose on our choice of codes is that the order
must be maintained, we can use any set of codes that are in order. For example, we can
also assign the following codes:
Poor 6, Fair 18, Good 23, Very good 45, Excellent 88
As we discuss in Chapter 19, which introduces statistical inference techniques for
ordinal data, the use of any code that preserves the order of the data will produce
exactly the same result. Thus, it’s not the magnitude of the values that is important, it’s
their order.
Students often have difficulty distinguishing between ordinal and interval data.
The critical difference between them is that the intervals or differences between values
of interval data are consistent and meaningful (which is why this type of data is called
interval ). For example, the difference between marks of 85 and 80 is the same five-mark
difference that exists between 75 and 70—that is, we can calculate the difference and
interpret the results.
Because the codes representing ordinal data are arbitrarily assigned except for
the order, we cannot calculate and interpret differences. For example, using a 1-2-34-5 coding system to represent poor, fair, good, very good, and excellent, we note
that the difference between excellent and very good is identical to the difference
between good and fair. With a 6-18-23-45-88 coding, the difference between excellent and very good is 43, and the difference between good and fair is 5. Because both
coding systems are valid, we cannot use either system to compute and interpret
differences.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
15
Here is another example. Suppose that you are given the following list of the most
active stocks traded on the NASDAQ in descending order of magnitude:
Order
Most Active Stocks
1
2
3
4
5
Microsoft
Cisco Systems
Dell Computer
Sun Microsystems
JDS Uniphase
Does this information allow you to conclude that the difference between the number of
stocks traded in Microsoft and Cisco Systems is the same as the difference in the number of stocks traded between Dell Computer and Sun Microsystems? The answer is
“no” because we have information only about the order of the numbers of trades, which
are ordinal, and not the numbers of trades themselves, which are interval. In other
words, the difference between 1 and 2 is not necessarily the same as the difference
between 3 and 4.
Calculations for Types of Data
Interval Data
All calculations are permitted on interval data. We often describe a set of interval data
by calculating the average. For example, the average of the 10 marks listed on page 13
is 70.3. As you will discover, there are several other important statistics that we will
introduce.
Nominal Data
Because the codes of nominal data are completely arbitrary, we cannot perform any calculations on these codes. To understand why, consider a survey that asks people to
report their marital status. Suppose that the first 10 people surveyed gave the following
responses:
single,
divorced
married,
married,
married,
widowed,
single,
married,
married,
single,
Using the codes
Single 1, married 2, divorced 3, widowed 4
we would record these responses as
1
2
2
2
4
1
2
2
1
3
The average of these numerical codes is 2.0. Does this mean that the average person is
married? Now suppose four more persons were interviewed, of whom three are widowed and one is divorced. The data are given here:
1
2
2
2
4
1
2
2
1
3
4
4
4
3
The average of these 14 codes is 2.5. Does this mean that the average person is married—but halfway to getting divorced? The answer to both questions is an emphatic
“no.” This example illustrates a fundamental truth about nominal data: Calculations
based on the codes used to store this type of data are meaningless. All that we are permitted to do with nominal data is count or compute the percentages of the occurrences of
each category. Thus, we would describe the 14 observations by counting the number of
each marital status category and reporting the frequency as shown in the following table.
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Category
Code
Frequency
Single
Married
Divorced
Widowed
1
2
3
4
3
5
2
4
The remainder of this chapter deals with nominal data only. In Chapter 3, we
introduce graphical techniques that are used to describe interval data.
Ordinal Data
The most important aspect of ordinal data is the order of the values. As a result, the
only permissible calculations are those involving a ranking process. For example, we can
place all the data in order and select the code that lies in the middle. As we discuss in
Chapter 4, this descriptive measurement is called the median.
Hierarchy of Data
The data types can be placed in order of the permissible calculations. At the top of the
list, we place the interval data type because virtually all computations are allowed. The
nominal data type is at the bottom because no calculations other than determining frequencies are permitted. (We are permitted to perform calculations using the frequencies of codes, but this differs from performing calculations on the codes themselves.) In
between interval and nominal data lies the ordinal data type. Permissible calculations
are ones that rank the data.
Higher-level data types may be treated as lower-level ones. For example, in universities and colleges, we convert the marks in a course, which are interval, to letter grades,
which are ordinal. Some graduate courses feature only a pass or fail designation. In this
case, the interval data are converted to nominal. It is important to point out that when
we convert higher-level data as lower-level we lose information. For example, a mark of 83
on an accounting course exam gives far more information about the performance of
that student than does a letter grade of A, which might be the letter grade for marks
between 80 and 90. As a result, we do not convert data unless it is necessary to do so.
We will discuss this later.
It is also important to note that we cannot treat lower-level data types as higherlevel types.
The definitions and hierarchy are summarized in the following box.
Types of Data
Interval
Values are real numbers.
All calculations are valid.
Data may be treated as ordinal or nominal.
Ordinal
Values must represent the ranked order of the data.
Calculations based on an ordering process are valid.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
17
Data may be treated as nominal but not as interval.
Nominal
Values are the arbitrary numbers that represent categories.
Only calculations based on the frequencies or percentages of occurrence
are valid.
Data may not be treated as ordinal or interval.
Interval, Ordinal, and Nominal Variables
The variables whose observations constitute our data will be given the same name as the
type of data. Thus, for example, interval data are the observations of an interval variable.
Problem Objectives and Information
In presenting the different types of data, we introduced a critical factor in deciding which
statistical procedure to use. A second factor is the type of information we need to produce from our data. We discuss the different types of information in greater detail in
Section 11.4 when we introduce problem objectives. However, in this part of the book
(Chapters 2–5), we will use statistical techniques to describe a set of data, compare two or
more sets of data, and describe the relationship between two variables. In Section 2.2, we
introduce graphical and tabular techniques employed to describe a set of nominal data.
Section 2.3 shows how to describe the relationship between two nominal variables and
compare two or more sets of nominal data.
EXERCISES
2.1 Provide two examples each of nominal, ordinal, and
interval data.
2.2 For each of the following examples of data, deter-
mine the type.
a. The number of miles joggers run per week
b. The starting salaries of graduates of MBA
programs
c. The months in which a firm’s employees choose
to take their vacations
d. The final letter grades received by students in a
statistics course
2.3 For each of the following examples of data, deter-
mine the type.
a. The weekly closing price of the stock of
Amazon.com
b. The month of highest vacancy rate at a La
Quinta motel
c. The size of soft drink (small, medium, or large)
ordered by a sample of McDonald’s customers
d. The number of Toyotas imported monthly by the
United States over the last 5 years
e. The marks achieved by the students in a statistics
course final exam marked out of 100
2.4 The placement office at a university regularly sur-
veys the graduates 1 year after graduation and asks
for the following information. For each, determine
the type of data.
a. What is your occupation?
b. What is your income?
c. What degree did you obtain?
d. What is the amount of your student loan?
e. How would you rate the quality of instruction?
(excellent, very good, good, fair, poor)
2.5 Residents of condominiums were recently surveyed
and asked a series of questions. Identify the type of
data for each question.
a. What is your age?
b. On what floor is your condominium?
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c. Do you own or rent?
d. How large is your condominium (in square feet)?
e. Does your condominium have a pool?
2.8 Baseball fans are regularly asked to offer their opin-
ions about various aspects of the sport. A survey asked
the following questions. Identify the type of data.
a. How many games do you attend annually?
b. How would you rate the quality of entertainment? (excellent, very good, good, fair, poor)
c. Do you have season tickets?
d. How would you rate the quality of the food? (edible, barely edible, horrible)
2.6 A sample of shoppers at a mall was asked the follow-
ing questions. Identify the type of data each question
would produce.
a. What is your age?
b. How much did you spend?
c. What is your marital status?
d. Rate the availability of parking: excellent, good,
fair, or poor
e. How many stores did you enter?
2.9 A survey of golfers asked the following questions.
Identify the type of data each question produces.
a. How many rounds of golf do you play annually?
b. Are you a member of a private club?
c. What brand of clubs do you own?
2.7 Information about a magazine’s readers is of interest
to both the publisher and the magazine’s advertisers.
A survey of readers asked respondents to complete
the following:
a. Age
b. Gender
c. Marital status
d. Number of magazine subscriptions
e. Annual income
f. Rate the quality of our magazine: excellent, good,
fair, or poor
2.10 At the end of the term, university and college stu-
For each item identify the resulting data type.
2 . 2 DESCRIBING
A
SET
OF
dents often complete questionnaires about their
courses. Suppose that in one university, students
were asked the following.
a. Rate the course (highly relevant, relevant, irrelevant)
b. Rate the professor (very effective, effective, not
too effective, not at all effective)
c. What was your midterm grade (A, B, C, D, F)?
Determine the type of data each question produces.
N O M I N A L D ATA
As we discussed in Section 2.1, the only allowable calculation on nominal data is to
count the frequency or compute the percentage that each value of the variable represents. We can summarize the data in a table, which presents the categories and their
counts, called a frequency distribution A relative frequency distribution lists the
categories and the proportion with which each occurs. We can use graphical techniques
to present a picture of the data. There are two graphical methods we can use: the bar
chart and the pie chart.
EXA MPLE 2.1
DATA
GSS2008*
Work Status in the GSS 2008 Survey
In Chapter 1, we briefly introduced the General Social Survey. In the 2008 survey
respondents were asked the following.
“Last week were you working full time, part time, going to school, keeping house,
or what”? The responses were
1. Working full-time
2. Working part-time
3. Temporarily not working
4. Unemployed, laid off
5. Retired
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19
GRAPHICAL DESCRIPTIVE TECHNIQUES I
6. School
7. Keeping house
8. Other
The responses were recorded using the codes 1, 2, 3, 4, 5, 6, 7, and 8, respectively. The
first 150 observations are listed here. The name of the variable is WRKSTAT, and the
data are stored in the 16th column (column P in Excel, column 16 in Minitab).
Construct a frequency and relative frequency distribution for these data and graphically summarize the data by producing a bar chart and a pie chart.
1
5
7
1
8
1
1
2
1
1
1
7
1
1
1
1
1
5
1
1
1
1
2
6
8
1
1
6
7
1
1
5
1
5
1
1
1
7
4
1
1
2
1
1
7
1
5
7
1
1
7
5
2
1
1
6
5
1
1
2
7
1
1
1
6
5
1
4
1
1
1
5
7
1
7
5
5
2
7
5
1
8
1
1
6
3
4
1
8
1
5
1
7
2
1
2
7
2
1
1
1
5
1
5
5
1
1
6
3
1
5
7
2
2
1
1
1
1
1
1
7
1
1
7
2
8
1
1
1
1
1
4
1
2
2
1
4
1
3
2
1
2
1
7
4
5
5
1
1
1
SOLUTION
Scan the data. Have you learned anything about the responses of these 150 Americans?
Unless you have special skills you have probably learned little about the numbers. If we
had listed all 2,023 observations you would be even less likely to discover anything useful about the data. To extract useful information requires the application of a statistical
or graphical technique. To choose the appropriate technique we must first identify the
type of data. In this example the data are nominal because the numbers represent categories. The only calculation permitted on nominal data is to count the number of
occurrences of each category. Hence, we count the number of 1s, 2s, 3s, 4s, 5s, 6s, 7s,
and 8s. The list of the categories and their counts constitute the frequency distribution.
The relative frequency distribution is produced by converting the frequencies into proportions. The frequency and relative frequency distributions are combined in Table 2.1.
TABLE
2 .1
Frequency and Relative Frequency Distributions for Example 2.1
WORK STATUS
Working full-time
CODE
FREQUENCY
RELATIVE FREQUENCY (%)
1
1003
49.6
Working part-time
2
211
10.4
Temporarily not working
3
53
2.6
Unemployed, laid off
4
74
3.7
Retired
5
336
16.6
School
6
57
2.8
Keeping house
7
227
11.2
Other
8
60
3.0
Total
2021
100
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There were two individuals who refused to answer hence the number of observations is
the sample size 2,023 minus 2, which equals 2,021.
As we promised in Chapter 1 (and the preface), we demonstrate the solution of all
examples in this book using three approaches (where feasible): manually, using Excel,
and using Minitab. For Excel and Minitab, we provide not only the printout but also
instructions to produce them.
EXCEL
INSTRUCTIONS
(Specific commands for this example are highlighted.)
1. Type or import the data into one or more columns. (Open GSS2008.)
2. Activate any empty cell and type
=COUNTIF ([Input range], [Criteria])
Input range are the cells containing the data. In this example, the range is P1:P2024.
The criteria are the codes you want to count: (1) (2) (3) (4) (5) (6) (7) (8). To count the
number of 1s (“Working full-time”), type
=COUNTIF (P1:P2024, 1)
and the frequency will appear in the dialog box. Change the criteria to produce the frequency of the other categories.
MINITAB
WRKSTAT
Count
Percent
1
2
3
4
5
6
7
8
1003
211
53
74
336
57
227
60
49.63
10.44
2.62
3.66
16.63
2.82
11.23
2.97
N 2021
* 2
INSTRUCTIONS
(Specific commands for this example are highlighted.)
1. Type or import the data into one column. (Open GSS2008.)
2. Click Stat, Tables, and Tally Individual Variables.
3. Type or use the Select button to specify the name of the variable or the column where
the data are stored in the Variables box (WRKSTAT). Under Display, click Counts
and Percents.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
21
INTERPRET
Almost 50% of respondents are working full-time, 16.6% are retired, 11.2% are keeping house, 10.4% are working part-time, and the remaining 12.1% are divided almost
equally among the other four categories.
Bar and Pie Charts
The information contained in the data is summarized well in the table. However, graphical
techniques generally catch a reader’s eye more quickly than does a table of numbers. Two
graphical techniques can be used to display the results shown in the table. A bar chart is
often used to display frequencies; a pie chart graphically shows relative frequencies.
The bar chart is created by drawing a rectangle representing each category. The
height of the rectangle represents the frequency. The base is arbitrary. Figure 2.1
depicts the manually drawn bar chart for Example 2.1.
FIGURE
2.1
Bar Chart for Example 2.1
1200
Frequency
1000
800
600
400
200
er
th
O
ho
ol
Ke
e
ho pin
us g
e
Sc
ed
tir
Re
W
fu ork
ll- in
tim g
e
W
pa o
rt- rki
tim ng
T
e
no em
t w po
r
o ar
U rkin ily
ne
g
m
pl
o
la ye
id d
of ,
f
0
If we wish to emphasize the relative frequencies instead of drawing the bar chart,
we draw a pie chart. A pie chart is simply a circle subdivided into slices that represent
the categories. It is drawn so that the size of each slice is proportional to the percentage
corresponding to that category. For example, because the entire circle is composed of
360 degrees, a category that contains 25% of the observations is represented by a slice
of the pie that contains 25% of 360 degrees, which is equal to 90 degrees. The number
of degrees for each category in Example 2.1 is shown in Table 2.2.
TABLE
2.2
Proportion in Each Category in Example 2.1
WORK STATUS
RELATIVE FREQUENCY (%)
SLICE OF THE PIE (º)
Working full-time
49.6
178.7
Working part-time
10.4
37.6
Temporarily not working
2.6
9.4
Unemployed, laid off
3.7
13.2
Retired
16.6
59.9
School
2.8
10.2
11.2
40.4
Other
3.0
10.7
Total
100.0
Keeping house
360
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CHAPTER 2
Figure 2.2 was drawn from these results.
FIGURE
2.2
Pie Chart for Example 2.1
Other
3%
Keeping house
11.2%
School
2.8%
Working full-time
49.6%
Retired
16.6%
Unemployed, laid off
3.7%
Temporarily not working
2.6%
Working part-time
10.4%
EXCEL
Here are Excel’s bar and pie charts.
1200
1003
8
3%
7
11%
1000
6
3%
800
600
400
5
17%
336
227
211
200
0
1
2
1
50%
53
74
3
4
60
57
5
6
7
4
4%
8
3
2%
2
10%
WRKSTAT
INSTRUCTIONS
1. After creating the frequency distribution, highlight the column of frequencies.
2. For a bar chart, click Insert, Column, and the first 2-D Column.
3. Click Chart Tools (if it does not appear, click inside the box containing the bar chart)
and Layout. This will allow you to make changes to the chart. We removed the
Gridlines, the Legend, and clicked the Data Labels to create the titles.
4. For a pie chart, click Pie and Chart Tools to edit the graph.
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23
GRAPHICAL DESCRIPTIVE TECHNIQUES I
MINITAB
1200
Pie Chart
1003
1000
8
*
3.0% 0.1
7
11.2%
Count
800
6
2.8%
600
400
227
211
1
49.6%
5
16.6%
336
200
0
1
2
53
74
3
4
60
57
5
6
WRKSTAT
7
8
2
*
4
3.7% 3
2.6%
Category
1
2
3
4
5
6
7
8
*
2
10.4%
INSTRUCTIONS
1. Type or import the data into one column. (Open GSS2008.)
For a bar chart:
2. Click Graph and Bar Chart.
3. In the Bars represent box, click Counts of unique values and select Simple.
4. Type or use the Select button to specify the variable in the Variables box (WRKSTAT).
We clicked Labels and added the title and clicked Data Labels and Use y-value labels
to display the frequencies at the top of the columns.
For a pie chart:
2. Click Graph and Pie Chart.
3. Click Chart, Counts of unique values, and in the Categorical variables box type or
use the Select button to specify the variable (WRKSTAT).
We clicked Labels and added the title. We clicked Slice Labels and clicked Category
name and Percent.
INTERPRET
The bar chart focuses on the frequencies and the pie chart focuses on the proportions.
Other Applications of Pie Charts and Bar Charts
Pie and bar charts are used widely in newspapers, magazines, and business and government reports. One reason for this appeal is that they are eye-catching and can attract
the reader’s interest whereas a table of numbers might not. Perhaps no one understands
this better than the newspaper USA Today, which typically has a colored graph on the
front page and others inside. Pie and bar charts are frequently used to simply present
numbers associated with categories. The only reason to use a bar or pie chart in such a
situation would be to enhance the reader’s ability to grasp the substance of the data. It
might, for example, allow the reader to more quickly recognize the relative sizes of the
categories, as in the breakdown of a budget. Similarly, treasurers might use pie charts to
show the breakdown of a firm’s revenues by department, or university students might
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CHAPTER 2
use pie charts to show the amount of time devoted to daily activities (e.g., eat 10%,
sleep 30%, and study statistics 60%).
APPLICATIONS in ECONOMICS
Macroeconomics
Macroeconomics is a major branch of economics that deals with the behavior of the economy as
a whole. Macroeconomists develop mathematical models that predict variables such as gross
domestic product, unemployment rates, and inflation. These are used by governments and corporations to help develop strategies. For example, central banks attempt to control inflation by lowering or raising interest rates. To do this requires that economists determine the effect of a
variety of variables, including the supply and demand for energy.
APPLICATIONS in ECONOMICS
Energy Economics
One variable that has had a large influence on the economies of virtually every
© Aaron Kohr/Shutterstock
country is energy. The 1973 oil crisis in which the price of oil quadrupled over a
short period of time is generally considered to be one of the largest financial
shocks to our economy. In fact, economists often refer to two different
economies: before the 1973 oil crisis and after.
Unfortunately, the world will be facing more shocks to our economy because
of energy for two primary reasons. The first is the depletion of nonrenewable sources
of energy and the resulting price increases. The second is the possibility that burning fossil
fuels and the creation of carbon dioxide may be the cause of global warming. One economist predicted that the cost of global warming will be calculated in the trillions of dollars. Statistics can
play an important role by determining whether Earth’s temperature has been increasing and, if so,
whether carbon dioxide is the cause. (See Case 3.1.)
In this chapter, you will encounter other examples and exercises that involve the issue of energy.
EXAMPLE
2.2
DATA
Xm02-02
Energy Consumption in the United States in 2007
Table 2.3 lists the total energy consumption of the United States from all sources in
2007 (latest data available at publication). To make it easier to see the details, the table
measures the energy in quadrillions of British thermal units (BTUs). Use an appropriate graphical technique to depict these figures.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
TABLE
2.3
25
Energy Consumption in the United States by Source, 2007
ENERGY SOURCES
QUADRILLIONS OF BTUS
Nonrenewable
Petroleum
39.773
Natural Gas
23.637
Coal and coal products
22.801
Nuclear
8.415
Renewable Energy Sources
Hydroelectric
2.446
Wood derived fuels
2.142
Biofuels
1.024
Waste
0.430
Geothermal
0.349
Wind
0.341
Solar/photovoltaic
0.081
Total
101.439
Sources: Non-renewable energy: Energy Information Administration (EIA), Monthly Energy Review (MER) December 2008, DOE/EIA-0035 (2008/12)
(Washington, DC: December 2008) Tables 1.3, 1.4a, and 1.4b; Renewable Energy: Table 1.2 of this report.
SOLUTION
We’re interested in describing the proportion of total energy consumption for each
source. Thus, the appropriate technique is the pie chart. The next step is to determine
the proportions and sizes of the pie slices from which the pie chart is drawn. The following pie chart was created by Excel. Minitab’s would be similar.
FIGURE
2.3
Pie Chart for Example 2.2
Biofuels
1.01%
Wood
2.11%
Hydroelectric
2.41%
Waste
0.42%
Wind
Solar 0.34%
0.08%
Geothermal
0.34%
Nuclear
8.30%
Coal and coal
products
22.48%
Petroleum
39.21%
Natural gas
23.30%
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INTERPRET
The United States depends heavily on petroleum, coal, and, natural gas. About 85% of
national energy use is based on these sources. The renewable energy sources amount to
less than 7%, of which about a third is hydroelectric and probably cannot be expanded
much further. Wind and solar barely appear in the chart.
See Exercises 2.11 to 2.15 for more information on the subject.
EXAMPLE
2.3
DATA
Xm02-03
Per Capita Beer Consumption (10 Selected Countries)
Table 2.4 lists the per capita beer consumption for each of 20 countries around the
world. Graphically present these numbers.
TABLE
2.4
Per Capita Beer Consumption 2008
COUNTRY
BEER CONSUMPTION(L/YR)
Australia
119.2
Austria
106.3
Belgium
93.0
Canada
68.3
Croatia
Czech Republic
Denmark
Finland
81.2
138.1
89.9
85.0
Germany
147.8
Hungary
75.3
Ireland
138.3
Luxembourg
84.4
Netherlands
79.0
New Zealand
77.0
Poland
69.1
Portugal
59.6
Slovakia
84.1
Spain
83.8
United Kingdom
96.8
United States
81.6
Source: www.beerinfo.com
SOLUTION
In this example, we’re primarily interested in the numbers. There is no use in presenting proportions here.
The following is Excel’s bar chart.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
2.4
EXCEL Bar Chart for Example 2.3
160
147.8
138.1
140
138.3
119.2
106.3
120
93.0
100
68.3
80
81.2
89.9
96.8
85.0
75.3
60
84.4
79.0 77.0
84.1 83.8
69.1
81.6
59.6
40
20
0
Au
s
tra
l
Au ia
st
ria
Be
lg
iu
m
Ca
na
d
a
Cz
ec Cro
h
a
Re tia
pu
bl
i
D
en c
m
a
Fi rk
nl
an
G
er d
m
a
H ny
un
ga
ry
Lu Irel
xe and
m
N bo
et
he urg
r
N
ew lan
d
Ze s
al
an
d
Po
la
nd
Po
rtu
Sl ga
ov l
ak
U
ia
ni
te
d Spa
Ki
i
n n
U
ni gdo
te
m
d
St
at
es
Beer Consumption (liters per capita)
FIGURE
27
Country
INTERPRET
Germany, the Czech Republic, Ireland, Australia, and Austria head the list. Both the
United States and Canada rank far lower. Surprised?
Describing Ordinal Data
There are no specific graphical techniques for ordinal data. Consequently, when we
wish to describe a set of ordinal data, we will treat the data as if they were nominal and
use the techniques described in this section. The only criterion is that the bars in bar
charts should be arranged in ascending (or descending) ordinal values; in pie charts, the
wedges are typically arranged clockwise in ascending or descending order.
We complete this section by describing when bar and pie charts are used to summarize and present data.
Factors That Identify When to Use Frequency and Relative Frequency
Tables, Bar and Pie Charts
1. Objective: Describe a single set of data.
2. Data type: Nominal or ordinal
EXERCISES
2.11 Xr02-11 When will the world run out of oil? One
way to judge is to determine the oil reserves of the
countries around the world. The next table displays
the known oil reserves of the top 15 countries.
Graphically describe the figures.
Country
Reserves
Brazil
Canada
China
12,620,000,000
178,100,000,000
16,000,000,000
(Continued)
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CHAPTER 2
Country
Iran
Iraq
Kazakhstan
Kuwait
Libya
Nigeria
Qatar
Russia
Saudi Arabia
United Arab Emirates
United States
Venezuela
Reserves
136,200,000,000
115,000,000,000
30,000,000,000
104,000,000,000
43,660,000,000
36,220,000,000
15,210,000,000
60,000,000,000
266,700,000,000
97,800,000,000
21,320,000,000
99,380,000,000
Marketable coke
Residual fuel oil
Liquefied refinery gas
Asphalt and road oil
Lubricants
Other
5.0
3.3
2.8
1.9
.9
1.5
Source: California Energy Commission based on 2004 data.
2.15 Xr02-15* The following table displays the energy con-
sumption pattern of Australia. The figures measure
the heat content in metric tons (1,000 kilograms) of
oil equivalent. Draw a graph that depicts these numbers and explain what you have learned.
Source: CIA World Factbook.
Energy Sources
2.12 Refer to Exercise 2.11. The total reserves in the world
are 1,348,528,420,000 barrels. The total reserves of
the top 15 countries are 1,232,210,000,000 barrels.
Use a graphical technique that emphasizes the
percentage breakdown of the top 15 countries plus
others. Briefly describe your findings.
Nonrenewable
Coal and coal products
Oil
Natural Gas
Nuclear
2.13 Xr02-13 The following table lists the average oil con-
Brazil
Canada
China
France
Germany
India
Iran
Italy
Japan
Mexico
Russia
Saudi Arabia
South Korea
United Kingdom
United States
Consumption
(barrels per day)
2,520,000
2,260,000
7,850,000
1,986,000
2,569,000
2,940,000
1,755,000
1,639,000
4,785,000
2,128,000
2,900,000
2,380,000
2,175,000
1,710,000
19,500,000
warming, which may be caused by the burning of fossil fuels (petroleum, natural gas, and coal) that produces carbon dioxide (CO2). The following table lists
the top 15 producers of CO2 and the annual amounts
(million of metric tons) from fossil fuels. Graphically
depict these figures. Explain what you have learned.
number of products produced and the proportion of
the total are listed in the following table. Draw a
graph to depict these numbers. What can you conclude from your graph?
Gasoline
Distillate fuel oil
Jet fuel
Still gas
51.4
15.3
12.6
5.4
347
115,396
2.16 Xr02-16 The planet may be threatened by global
2.14 Xr02-14 There are 42 gallons in a barrel of oil. The
Percent of Total (%)
1,388
4,741
Source: International Energy Association.
Source: CIA World Factbook.
Product
55,385
33,185
20,350
0
Renewable Energy Sources
Hydroelectric
Solid Biomass
Other (Liquid biomass,
geothermal, solar, wind,
and tide, wave, and ocean)
Total
sumption per day for the top 15 oil-consuming
countries. Use a graphical technique to present
these figures.
Country
Heat Content
Country
CO2
Australia
Canada
China
France
Germany
India
Iran
Italy
406.6
631.3
5322.7
415.3
844.2
1165.7
450.7
466.6
Country
CO2
Japan
1230.4
Korea, South
499.6
Russia
1696.0
Saudi Arabia
412.4
South Africa
423.8
United Kingdom 577.2
United States
5957.0
Source: Statistical Abstract of the United States, 2009, Table 1304.
2.17
The production of steel has often been used
as a measure of the economic strength of a country.
The following table lists the steel production in the
20 largest steel-producing nations in 2008. The
units are millions of metric tons. Use a graphical
technique to display these figures.
Xr02-17
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29
GRAPHICAL DESCRIPTIVE TECHNIQUES I
Country
Steel
production
Belgium
Brazil
Canada
China
France
Germany
India
Iran
Italy
Japan
10.7
33.7
14.8
500.5
17.9
45.8
55.2
10
30.6
118.7
Country
Steel
production
Mexico
Poland
Russia
South Korea
Spain
Taiwan
Turkey
Ukraine
United Kingdom
United States
17.2
9.7
68.5
53.6
18.6
19.9
26.8
37.1
13.5
91.4
Source: World Steel Association.
Recycled Organic/
Yard
10%
Recycled Other
4%
Non-Residential
Recycled
Pct
Recycled Glass
1%
Recycled Paper
11%
Recycled Metal
3%
Recycled Organic
1%
Recycled Construction/Demolition 1%
Recycled Other
1%
2.18 Xr02-18 In 2003 (latest figures available) the United
States generated 251.3 million tons of garbage. The
following table lists the amounts by source. Use one
or more graphical techniques to present these figures.
Amount
(millions of tons)
Source
Paper and paperboard
Glass
Metals
Plastics
Rubber and leather
Textiles
Wood
Food scraps
Yard trimmings
Other
2.19 Xr02-19 In the last five years, the city of Toronto has
intensified its efforts to reduce the amount of
garbage that is taken to landfill sites. [Currently, the
Greater Toronto Area (GTA) disposes of its garbage
in a dump site in Michigan.] A current analysis of
GTA reveals that 36% of waste collected is taken
from residences and 64% from businesses and public
institutions (hospitals, schools, universities, etc.). A
further breakdown is listed below. (Source: Toronto
City Summit Alliance.)
a. Draw a pie chart for residential waste including
both recycled and disposed waste.
b. Repeat part (a) for nonresidential waste.
Residential
Pct
Recycled Plastic
1%
Recycled Glass
3%
Recycled Paper
14%
Recycled Metal
1%
Recycled Organic/
Food
7%
Plastic
10%
Glass
3%
Paper
31%
Metal
8%
Organic
18%
Construction
/Demolition 7%
Other
6%
and amounts of oil (millions of barrels annually)
they exported to the United States in 2007.
Country
Oil Imports
162
181
681
177
64
514
395
530
37
420
Source: Statistical Abstract of the United States, 2009, Table 895.
Source: Statistical Abstract of the United States, 2009, Table 361.
Recycled
Pct
2.20 Xr02-20 The following table lists the top 10 countries
Algeria
Angola
Canada
Iraq
Kuwait
Mexico
Nigeria
Saudi Arabia
United Kingdom
Venezuela
85.2
13.3
19.1
29.4
6.5
11.8
13.8
31.2
32.4
8.6
Disposed
Disposed
Pct
Plastic
Paper
Metal
Organic
Other
7%
12%
2%
23%
17%
a. Draw a bar chart.
b. Draw a pie chart.
c. What information is conveyed by each chart?
2.21 Xr02-21 The following table lists the percentage of males
and females in five age groups that did not have health
insurance in the United States in September 2008. Use
a graphical technique to present these figures.
Age Group
Male
Female
Under 18
18–24
25–34
35–44
45–64
8.5
32.3
30.4
21.3
13.5
8.5
24.9
21.4
17.1
13.0
Source: National Health Interview Survey.
2.22 Xr02-22 The following table lists the average costs
for a family of four to attend a game at a National
Football League (NFL) stadium compared to a
Canadian Football League (CFL) stadium. Use a
graphical technique that allows the reader to compare each component of the total cost.
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CHAPTER 2
Four tickets
Parking
Two ball caps
Two beers
Two drinks
Four hot dogs
NFL
CFL
274.12
19.75
31.12
11.90
7.04
15.00
171.16
10.85
44.26
11.24
7.28
16.12
Source: Team Market Report, Matthew Coutts.
2.23 Xr02-23 Productivity growth is critical to the eco-
nomic well-being of companies and countries. In the
table below we list the average annual growth rate
(in percent) in productivity for the Organization for
Economic Co-Operation and Development
(OECD) countries. Use graphical technique to present these figures.
Country
Australia
Austria
Belgium
Canada
Czech
Republic
Denmark
Finland
France
Germany
Greece
Hungary
Iceland
Ireland
Italy
Productivity
Growth
1.6
1.5
1.975
1.25
3.3
2.175
2.775
2.35
2.3
1.6
3.6
0.775
3.775
1.525
Country
Productivity
Growth
Japan
Korea
Luxembourg
Mexico
Netherlands
New Zealand
Norway
Portugal
Slovak Republic
Spain
Sweden
Switzerland
United
Kingdom
United
States
2.775
5.55
2.6
1.2
2
1.5
2.575
2.8
4.8
2
1.775
1.033
2.2
1.525
Source: OECD Labor Productivity Database July 2007.
The following exercises require a computer and software.
2.24 Xr02-24 In an attempt to stimulate the economy in
2008, the U.S. government issued rebate checks
totaling $107 billion. A survey conducted by the
National Retail Federation (NRF) asked recipients
what they intended to do with their rebates. The
choices are:
1.
2.
3.
4.
5.
6.
Buy something
Pay down debt
Invest
Pay medical bills
Save
Other
Use a graphical technique to summarize and present
these data. Briefly describe your findings.
2.25 Xr02-25 Refer to Exercise 2.24. Those who respo-
nded that they planned to buy something were asked
what they intended to buy. Here is a list of their
responses.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Home improvement project
Purchase appliances
Purchase automobles
Purchase clothing
Purchase electronics
Purchase furniture
Purchase gas
Spa or salon time
Purchase vacation
Purchase groceries
Impulse purchase
Down payment on house
Graphically summarize these data. What can you
conclude from the chart?
2.26 Xr02-26 What are the most important characteristics
of colleges and universities? This question was asked
of a sample of college-bound high school seniors.
The responses are:
1.
2.
3.
4.
5.
6.
Location
Majors
Academic reputation
Career focus
Community
Number of students
The results are stored using the codes. Use a graphical technique to summarize and present the data.
2.27 Xr02-27 Where do consumers get information about
cars? A sample of recent car buyers was asked to
identify the most useful source of information about
the cars they purchased. The responses are:
1.
2.
3.
4.
Consumer guide
Dealership
Word of mouth
Internet
The responses were stored using the codes.
Graphically depict the responses. Source: Automotive
Retailing Today, The Gallup Organization.
2.28 Xr02-28 A survey asked 392 homeowners which area
of their homes they would most like to renovate.
The responses and frequencies are shown next. Use
a graphical technique to present these results.
Briefly summarize your findings.
Area
Basement
Bathroom
Bedroom
Kitchen
Living/dining room
Code
1
2
3
4
5
Source: Toronto Star, November 23, 2004.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
2.29 Xr02-29 Subway train riders frequently pass the time
by reading a newspaper. New York City has a subway and four newspapers. A sample of 360 subway
riders who regularly read a newspaper was asked to
identify that newspaper. The responses are:
1.
2.
3.
4.
New York Daily News
New York Post
New York Times
Wall Street Journal
The responses were recorded using the numerical
codes shown.
a. Produce a frequency distribution and a relative
frequency distribution.
b. Draw an appropriate graph to summarize the
data. What does the graph tell you?
2.30 Xr02-30 Who applies to MBA programs? To help
determine the background of the applicants, a sample of 230 applicants to a university’s business school
was asked to report their undergraduate degrees.
The degrees were recorded using these codes.
1.
2.
3
4.
5.
a.
b.
c.
d.
BA
BBA
BEng
BSc
Other
Determine the frequency distribution.
Draw a bar chart.
Draw a pie chart.
What do the charts tell you about the sample of
MBA applicants?
2.31 Xr02-31 Many business and economics courses
require the use of computer, so students often must
buy their own computers. A survey asks students to
identify which computer brands they have purchased. The responses are:
1.
2.
3.
4.
IBM
Compaq
Dell
Other
31
a. Use a graphical technique that depicts the frequencies.
b. Graphically depict the proportions.
c. What do the charts tell you about the brands of
computers used by the students?
2.32 Xr02-32 An increasing number of statistics courses
use a computer and software rather than manual calculations. A survey of statistics instructors asked
them to report the software their courses use. The
responses are:
1. Excel
2. Minitab
3. SAS
4. SPSS
5. Other
a. Produce a frequency distribution.
b. Graphically summarize the data so that the proportions are depicted.
c. What do the charts tell you about the software
choices?
2.33 Xr02-33* The total light beer sales in the United States
are approximately 3 million gallons annually. With
this large of a market, breweries often need to know
more about who is buying their product. The marketing manager of a major brewery wanted to analyze
the light beer sales among college and university students who drink light beer. A random sample of 285
graduating students was asked to report which of the
following is their favorite light beer:
1.
2.
3.
4.
5.
6.
7.
Bud Light
Busch Light
Coors Light
Michelob Light
Miller Lite
Natural Light
Other brands
The responses were recorded using the codes 1, 2, 3, 4,
5, 6, and 7, respectively. Use a graphical to summarize
these data. What can you conclude from the chart?
GENER AL SO CIAL SURVEY EXERCISES
The following exercises are based on the GSS described above.
2.34
In the 2008 General Social Survey, respondents were asked to identify their race (RACE) using
the following categories:
GSS2008*
1. White
2. Black
3. Other
Summarize the results using an appropriate graphical technique and interpret your findings.
2.35 GSS2008* Several questions deal with education. One
question in the 2008 survey asked respondents to
indicate their highest degree (DEGREE). The
responses are:
0.
1.
2.
3.
4.
Left high school
Completed high school
Completed junior college
Completed bachelor’s degree
Completed graduate degree
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CHAPTER 2
Use a graphical technique to summarize the data.
Describe what the graph tells you.
2.36 GSS2006* Refer to the GSS in 2006. Responses to the
question about marital status (MARITAL) were:
1.
2.
3.
4.
5.
Married
Widowed
Divorced
Separated
Never Married
2.37 GSS2004* Refer to the 2004 GSS, which asked about
individual’s class (CLASS). The responses were:
1.
2.
3.
4.
Lower class
Working class
Middle class
Upper class
Summarize the data using a graphical method and
describe your findings.
a. Create a frequency distribution
b. Use a graphical method to present these data and
briefly explain what the graph reveals.
2 . 3 D E S C R I B I N G T H E R E L AT I O N S H I P B E T W E E N T WO N O M I N A L
VA R I A B L E S A N D C O M PA R I N G T WO OR M OR E N O M I N A L
D ATA S E T S
In Section 2.2, we presented graphical and tabular techniques used to summarize a set of
nominal data. Techniques applied to single sets of data are called univariate. There are
many situations where we wish to depict the relationship between variables; in such cases,
bivariate methods are required. A cross-classification table (also called a cross-tabulation table) is used to describe the relationship between two nominal variables. A variation
of the bar chart introduced in Section 2.2 is employed to graphically describe the relationship. The same technique is used to compare two or more sets of nominal data.
Tabular Method of Describing the Relationship between Two
Nominal Variables
To describe the relationship between two nominal variables, we must remember that we
are permitted only to determine the frequency of the values. As a first step, we need to
produce a cross-classification table that lists the frequency of each combination of the
values of the two variables.
EXAMPLE
2.4
DATA
Xm02-04
Newspaper Readership Survey
A major North American city has four competing newspapers: the Globe and Mail
(G&M), Post, Star and Sun. To help design advertising campaigns, the advertising managers of the newspapers need to know which segments of the newspaper market are
reading their papers. A survey was conducted to analyze the relationship between newspapers read and occupation. A sample of newspaper readers was asked to report which
newspaper they read—Globe and Mail (1), Post (2), Star (3), Sun (4)—and indicate
whether they were blue-collar workers (1), white-collar workers (2), or professionals
(3). Some of the data are listed here.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
Reader
Occupation
Newspaper
2
1
2
.
.
3
1
2
2
4
1
.
.
2
3
3
1
2
3
.
.
352
353
354
33
Determine whether the two nominal variables are related.
SOLUTION
By counting the number of times each of the 12 combinations occurs, we produced the
Table 2.5.
TABLE
2.5
Cross-Classification Table of Frequencies for Example 2.4
NEWSPAPER
G&M
POST
STAR
SUN
TOTAL
Blue collar
27
18
38
37
120
White collar
29
43
21
15
108
Professional
33
51
22
20
126
Total
89
112
81
72
354
OCCUPATION
If occupation and newspaper are related, there will be differences in the newspapers
read among the occupations. An easy way to see this is to convert the frequencies in
each row (or column) to relative frequencies in each row (or column). That is, compute
the row (or column) totals and divide each frequency by its row (or column) total, as
shown in Table 2.6. Totals may not equal 1 because of rounding.
TABLE
2 .6
Row Relative Frequencies for Example 2.4
NEWSPAPER
G&M
POST
STAR
SUN
TOTAL
Blue collar
.23
.15
.32
.31
1.00
White collar
.27
.40
.19
.14
1.00
Professional
.26
.40
.17
.16
1.00
Total
.25
.32
.23
.20
1.00
OCCUPATION
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CHAPTER 2
EXCEL
Excel can produce the cross-classification table using several methods. We will use and
describe the PivotTable in two ways: (1) to create the cross-classification table featuring
the counts and (2) to produce a table showing the row relative frequencies.
Count of Reader
Newspaper
Occupation
Blue collar
White collar
Professional
Grand Total
G&M
27
29
33
89
Count of Reader
Post
18
43
51
112
Star
38
21
22
81
Sun
37
15
20
72
Grand Total
120
108
126
354
Post
0.15
0.40
0.40
0.32
Star
0.32
0.19
0.17
0.23
Sun
0.31
0.14
0.16
0.20
Grand Total
1.00
1.00
1.00
1.00
Newspaper
Occupation
Blue collar
White collar
Professional
Grand Total
G&M
0.23
0.27
0.26
0.25
INSTRUCTIONS
The data must be stored in (at least) three columns as we have done in Xm02-04. Put the
cursor somewhere in the data range.
1. Click Insert and PivotTable.
2. Make sure that the Table/Range is correct.
3. Drag the Occupation button to the ROW section of the box. Drag the Newspaper
button to the COLUMN section. Drag the Reader button to the DATA field. Rightclick any number in the table, click Summarize Data By, and check Count. To convert to row percentages, right-click any number, click Summarize Data By, More
options . . . , and Show values as. Scroll down and click % of rows. (We then formatted the data into decimals.) To improve both tables, we substituted the names of
the occupations and newspapers.
MINITAB
Tabulated statistics: Occupation, Newspaper
Rows: Occupation
1
2
3
All
Cell Contents:
1
27
22.50
29
26.85
33
26.19
89
25.14
Columns: Newspaper
2
18
15.00
43
39.81
51
40.48
112
31.64
3
38
31.67
21
19.44
22
17.46
81
22.88
4
37
30.83
15
13.89
20
15.87
72
20.34
All
120
100.00
108
100.00
126
100.00
354
100.00
Count
% of Row
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
35
INSTRUCTIONS
1. Type or import the data into two columns. (Open xM02-04)
2. Click Stat, Tables, and Cross Tabulation and Chi-square.
3. Type or use the Select button to specify the Categorical variables: For rows
(Occupation) and For columns (Newspaper)
4. Under Display, click Counts and Row percents (or any you wish)
INTERPRET
Notice that the relative frequencies in the second and third rows are similar and that there
are large differences between row 1 and rows 2 and 3. This tells us that blue-collar workers
tend to read different newspapers from both white-collar workers and professionals and
that white-collar workers and professionals are quite similar in their newspaper choices.
Graphing The Relationship between Two Nominal Variables
We have chosen to draw three bar charts, one for each occupation depicting the four
newspapers. We’ll use Excel and Minitab for this purpose. The manually drawn charts
are identical.
EXCEL
There are several ways to graphically display the relationship between two nominal
variables. We have chosen two dimensional bar charts for each of the three occupations.
The charts can be created from the output of the PivotTable (either counts as we have
done) or row proportions.
60
50
Star Sun
30
G&M
20
Post
Post
40
Post
G&M
G&M
Star
Sun
Star Sun
10
0
Blue collar
White collar
Professional
Occupation
INSTRUCTIONS
From the cross-classification table, click Insert and Column. You can do the same
from any completed cross-classification table.
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MINITAB
Minitab can draw bar charts from the raw data.
Chart of Occupation, Newspaper
50
Count
40
30
20
10
0
Newspaper
Occupation
1
2
3
4
1
2
1
3
4
1
2
2
3
4
3
INSTRUCTIONS
1. Click Graph and Bar Chart.
2. In the Bars represent box, specify Counts of unique values. Select Cluster.
3. In the Categorical variables box, type or select the two variables (Newspaper
Occupation).
If you or someone else has created the cross-classification table, Minitab can draw bar
charts directly from the table.
INSTRUCTIONS
1. Start with a completed cross-classification table such as Table 2.9.
2. Click Graph and Bar Chart
3. In the Bars represent box click Values from a table. Choose Two-way table
Cluster.
4. In the Graph variables box, Select the columns of numbers in the table. In the
Row labels box, Select the column with the categories.
Chart of Occupation, Newspaper
50
Count
40
30
20
10
0
Newspaper
Occupation
1
2
3
1
4
1
2
3
2
4
1
2
3
4
3
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
37
INTERPRET
If the two variables are unrelated, then the patterns exhibited in the bar charts should be
approximately the same. If some relationship exists, then some bar charts will differ
from others.
The graphs tell us the same story as did the table. The shapes of the bar charts for
occupations 2 and 3 (white-collar and professional) are very similar. Both differ considerably from the bar chart for occupation 1 (blue-collar).
Comparing Two or More Sets of Nominal Data
We can interpret the results of the cross-classification table of the bar charts in a different way. In Example 2.4, we can consider the three occupations as defining three
different populations. If differences exist between the columns of the frequency distributions (or between the bar charts), then we can conclude that differences exist
among the three populations. Alternatively, we can consider the readership of the
four newspapers as four different populations. If differences exist among the frequencies or the bar charts, then we conclude that there are differences between the four
populations.
DATA
ANES2008*
Using the technique introduced above, we produced the bar charts below.
©AP Photo/David Smith
Do Male and Female American Voters Differ in
Their Party Affiliation?
EXCEL
0.45
Axis Title
0.35
0.30
0.25
Dem
Ind
0.40
Dem
Ind
Rep
Rep
0.20
0.15
No
0.10
0.05
Other
No
(blank)
Other
(blank)
0.00
Male
Female
Gender
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CHAPTER 2
MINITAB
Chart of Gender, Party
400
Count
300
200
100
0
Party
Gender
1
2
3
4
1
5
*
1
2
3
4
5
*
2
INTERPRET
As you can see, there are substantial differences between the bar charts for men and women. We can conclude
that gender and party affiliation are related. However, we can also conclude that differences in party affiliation
exist between American male and female voters: Specifically, men tend to identify themselves as independents,
whereas women support the Democratic party.
Historically, women tend to be Democrats, and men lean toward the Republican party. However, in this
survey, both genders support the Democrats over the Republicans, which explains the results of the 2008
election.
Data Formats
There are several ways to store the data to be used in this section to produce a table or
a bar or pie chart.
1.
The data are in two columns. The first column represents the categories of the
first nominal variable, and the second column stores the categories for the second
variable. Each row represents one observation of the two variables. The number
of observations in each column must be the same. Excel and Minitab can produce
a cross-classification table from these data. (To use Excel’s PivotTable, there also
must be a third variable representing the observation number.) This is the way
the data for Example 2.4 were stored.
2.
The data are stored in two or more columns, with each column representing the
same variable in a different sample or population. For example, the variable may
be the type of undergraduate degree of applicants to an MBA program, and there
may be five universities we wish to compare. To produce a cross-classification
table, we would have to count the number of observations of each category
(undergraduate degree) in each column.
3.
The table representing counts in a cross-classification table may have already
been created.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
39
We complete this section with the factors that identify the use of the techniques
introduced here.
Factors That Identify When to Use a Cross-Classification Table
1. Objective: Describe the relationship between two variables and
compare two or more sets of data.
2. Data type: Nominal
EXERCISES
2.38
Xr02-38 Has the educational level of adults changed
over the past 15 years? To help answer this question,
the Bureau of Labor Statistics compiled the following table; it lists the number (1,000) of adults 25
years of age and older who are employed. Use a
graphical technique to present these figures. Briefly
describe what the chart tells you.
2.40
The table below displays the energy consumption patterns of Australia and New Zealand.
The figures measure the heat content in metric tons
(1,000 kilograms) of oil equivalent. Use a graphical
technique to display the differences between the
sources of energy for the two countries.
Xr02-15*
Energy Sources
Educational level
1995 1999
2003
2007
Less than high school
12,021 12,110
12,646
12,408
High school
36,746 35,335
33,792
32,634
Some college
30,908 30,401
30,338
30,389
College graduate
31,176 33,651
35,454
37,321
Nonrenewable
Coal & coal products
Oil
Natural Gas
Nuclear
Renewable
Hydroelectric
Solid Biomass
Other (Liquid biomass,
geothermal, solar, wind,
and tide, wave, &
ocean)
Total
Source: Statistical Abstract of the United States, 2009, Table 572.
2.39
How do governments spend the tax dollars
they collect, and has this changed over the past 15
years? The following table displays the amounts
spent by the federal, state, and local governments on
consumption expenditures and gross investments.
Consumption expenditures are services (such as
education). Gross investments ($billions) consist of
expenditures on fixed assets (such as roads, bridges,
and highways). Use a graphical technique to present
these figures. Have the ways governments spend
money changed over the previous 15 years?
Xr02-39
Level of Government
and Type
1990 1995
2000
2004
Federal national defense
Consumption
308.1
Gross
65.9
297.3
51.4
321.5
48.8
477.5
70.4
Federal nondefense
Consumption
Gross
111.7
22.6
143.2
27.3
177.8
30.7
227.0
35.0
State and local
Consumption
Gross
544.6
127.2
696.1
154.0
917.8
225.0
1,099.7
274.3
Source: Statistical Abstract of the United States, 2006, Table 419.
Australia
New Zealand
55,385
33,185
20,350
0
1,281
6,275
5,324
0
1,388
4,741
1,848
805
347
115,396
2,761
18,294
Source: International Energy Association.
The following exercises require a computer and software.
2.41
The average loss from a robbery in the United
States in 2004 was $1,308 (Source: U.S. Federal Bureau
of Investigation). Suppose that a government agency
wanted to know whether the type of robbery differed
between 1990, 1995, 2000, and 2006. A random sample
of robbery reports was taken from each of these years,
and the types were recorded using the codes below.
Determine whether there are differences in the types
of robbery over the 16-year span. (Adapted from
Statistical Abstract of the United States, 2009, Table 308.)
1. Street or highway
2. Commercial house
3. Gas station
4. Convenience store
5. Residence
6. Bank
7. Other
Xr02-41
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2.43
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CHAPTER 2
Xr02-42 The associate dean of a business school was
looking for ways to improve the quality of the applicants to its MBA program. In particular, she wanted
to know whether the undergraduate degree of applicants differed among her school and the three nearby
universities with MBA programs. She sampled 100
applicants of her program and an equal number from
each of the other universities. She recorded their
undergraduate degrees (1 , 2 BEng, 3 BBA, 4 other) as well the university (codes 1, 2, 3,
and 4). Use a tabular technique to determine
whether the undergraduate degree and the university
each person applied to appear to be related.
Is there brand loyalty among car owners in
their purchases of gasoline? To help answer the question, a random sample of car owners was asked to
record the brand of gasoline in their last two purchases
(1 Exxon, 2 Amoco, 3 Texaco, 4 Other). Use
a tabular technique to formulate your answer.
2.45
1.
2.
3.
4.
2.46
The costs of smoking for individuals, companies for whom they work, and society in general
is in the many billions of dollars. In an effort to
reduce smoking, various government and nongovernment organizations have undertaken information campaigns about the dangers of smoking.
Most of these have been directed at young people.
This raises the question: Are you more likely to
smoke if your parents smoke? To shed light on the
issue, a sample of 20- to 40-year-old people were
asked whether they smoked and whether their parents smoked. The results are stored the following
way:
Xr02-44
Use a tabular technique to produce the information
you need.
Lost job
Left job
Reentrants
New entrants
Determine whether there are differences between
unemployed men and women in terms of the reasons
for unemployment. (Source: Adapted from Statistical
Abstract of the United States, 2009 Table 604.)
Xr02-43
Column 1: 1 do not smoke, 2 smoke
Column 2: 1 neither parent smoked,
2 father smoked, 3 mother smoked,
4 both parents smoked
Xr02-45 In 2007, 3,882,000 men and 3,196,000
women were unemployed at some time during the
year (Source: U.S. Bureau of Labor Statistics). A statistics practitioner wanted to investigate the reasons
for unemployment and whether the reasons differed
by gender. A random sample of people 16 years of
age and older was drawn. The reasons given for
their status are:
In 2004, the total number of prescriptions
sold in the United States was 3,274,000,000 (Source:
National Association of Drug Store Chains). The
sales manager of a chain of drugstores wanted to
determine whether changes were made in the way
the prescriptions were filled. A survey of prescriptions was undertaken in 1995, 2000, and 2007. The
year and type of each prescription were recorded
using the codes below. Determine whether there are
differences between the years. (Source: Adapted from
the Statistical Abstract of the United States, 2009,
Table 151.)
Xr02-46
1.
2.
3.
4.
5.
2.47
Traditional chain store
Independent drugstore
Mass merchant
Supermarket
Mail order
Refer to Exercise 2.33. Also recorded was
the gender of the respondents. Use a graphical technique to determine whether the choice of light beers
differs between genders.
Xr02-33*
CHAPTER SUM M ARY
Descriptive statistical methods are used to summarize data
sets so that we can extract the relevant information. In this
chapter, we presented graphical techniques for nominal
data.
Bar charts, pie charts, and frequency distributions are
employed to summarize single sets of nominal data.
Because of the restrictions applied to this type of data, all
that we can show is the frequency and proportion of each
category.
To describe the relationship between two nominal
variables, we produce cross classification tables and bar
charts.
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GRAPHICAL DESCRIPTIVE TECHNIQUES I
41
IMPORTANT TERMS
Ordinal 14
Frequency distribution 18
Relative frequency
distribution 18
Bar chart 18
Pie chart 18
Univariate 32
Bivariate 32
Cross-classification table 32
Cross-tabulation table 32
Variable 13
Values 13
Data 13
Datum 13
Interval 14
Quantitative 14
Numerical 14
Nominal 14
Qualitative 14
Categorical 14
COMPUTER OUTPUT AND INSTRUCTIONS
Graphical Technique
Excel
Minitab
Bar chart
Pie chart
22
22
23
23
CHAPTER EXERCISES
1.
2.
3.
4.
The following exercises require a computer and software.
2.48
A sample of 200 people who had purchased
food at the concession stand at Yankee Stadium
was asked to rate the quality of the food. The
responses are:
Xr02-48
a.
b.
c.
d.
e.
Poor
Fair
Good
Very good
Excellent
Use a graphical technique that compares the proportions of Americans’ and Canadians’ use of search
engines
2.51
Draw a graph that describes the data. What does the
graph tell you?
2.49
a. Emphasize manual calculations
b. Use a computer combined with manual calculations
c. Use a computer exclusively with no manual calculations
2.50
Which Internet search engines are the most
popular? A survey undertaken by the Financial Post
(May 14, 2004) asked random samples of
Americans and Canadians that question. The
responses were:
Xr02-50
The Wilfrid Laurier University bookstore
conducts annual surveys of its customers. One question asks respondents to rate the prices of textbooks.
The wording is, “The bookstore’s prices of textbooks are reasonable.” The responses are:
Xr02-51
1.
2.
3.
4.
5.
There are several ways to teach applied statistics. The most popular approaches are:
Xr02-49
A survey of 100 statistics instructors asked each one
to report his or her approach. Use a graphical
method to extract the most useful information about
the teaching approaches.
Google
Microsoft (MSN)
Yahoo
Other
Strongly disagree
Disagree
Neither agree nor disagree
Agree
Strongly agree
The responses for a group of 115 students were
recorded. Graphically summarize these data and
report your findings.
2.52
The Red Lobster restaurant chain conducts
regular surveys of its customers to monitor the performance of individual restaurants. One question asks
customers to rate the overall quality of their last visit.
The listed responses are poor (1), fair (2), good (3),
very good (4), and excellent (5). The survey also asks
respondents whether their children accompanied
them to the restaurant (1 yes, 2 no). Graphically
depict these data and describe your findings.
Xr02-52
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CHAPTER 2
Xr02-53 Many countries are lowering taxes on corporations in an effort to make their countries more
attractive for investment. In the next table, we list
the marginal effective corporate tax rates among
Organization for Economic Co-Operation and
Development (OECD) countries. Develop a graph
that depicts these figures. Briefly describe your
results.
Slovak
Spain
Sweden
Switzerland
Turkey
United Kingdom
United States
2.54
Country
Manufacturers
Australia
Austria
Belgium
Canada
Czech Republic
Denmark
Finland
France
Germany
Greece
Hungary
Iceland
Ireland
Italy
Japan
Korea
Luxembourg
Mexico
Netherlands
New Zealand
Norway
Poland
Portugal
27.7
21.6
6.0
20.0
1.0
16.5
22.4
33.0
30.8
18.0
12.9
19.5
12.7
24.6
35.2
32.8
24.1
17.1
18.3
27.1
25.8
14.4
14.8
Services Aggregate
26.6
19.5
4.1
29.2
7.8
12.7
22.9
31.7
29.4
13.2
12.0
17.6
11.7
28.6
30.4
31.0
20.3
12.1
15.0
25.4
23.2
15.0
16.1
26.7
19.9
0.5
25.2
8.4
13.4
22.8
31.9
29.7
13.8
12.2
17.9
12.0
27.8
31.3
31.5
20.6
13.1
15.5
25.7
23.5
14.9
15.9
13.3
27.2
19.3
14.8
22.7
22.7
32.7
11.7
25.2
17.5
15.0
20.2
27.8
39.9
12.0
25.5
17.8
14.9
20.8
26.9
36.9
A survey of the business school graduates
undertaken by a university placement office asked,
among other questions, the area in which each
person was employed. The areas of employment
are:
Xr02-54*
a.
b.
c.
d.
e.
Accounting
Finance
General management
Marketing/Sales
Other
Additional questions were asked and the responses
were recorded in the following way.
Column
Variable
1
2
3
4
Identification number
Area
Gender (1 female, 2 male)
Job satisfaction (4 very, 3 quite,
2 little, 1 none)
The placement office wants to know the following:
a. Do female and male graduates differ in their
areas of employment? If so, how?
b. Are area of employment and job satisfaction
related?
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3
© Tonis Valling/Shutterstock
GRAPHICAL DESCRIPTIVE
TECHNIQUES II
Graphical Techniques to Describe a Set of Interval Data
3.2
Describing Time-Series Data
3.3
Describing the Relationship between Two Interval Variables
3.4
Art and Science of Graphical Presentations
Were Oil Companies Gouging Customers 2000–2009?
DATA
Xm03-00
The price of oil has been increasing for several reasons. First, oil is a finite resource; the
world will eventually run out. In January 2009, the world was consuming more than
100 million barrels per day—more than 36 billion barrels per year. The total proven
world reserves of oil are 1,348.5 billion barrels. At today’s consumption levels, the proven reserves
will be exhausted in 37 years. (It should be noted, however, that in 2005 the proven reserves of oil
© Comstock Images/Jupiterimages
3.1
amounted to 1,349.4 billion barrels, indicating that new oil discoveries are offsetting increasing usage.)
Second, China’s and India’s industries are rapidly increasing and require ever-increasing amounts of oil.
Third, over the last 10 years, hurricanes have threatened the oil rigs in the Gulf of Mexico.
On page 78 you will
find our answer
The result of the price increases in oil is reflected in the price of gasoline. In January 2000,
the average retail price of gasoline in the United States was $1.301 per U.S. gallon (one U.S.
43
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CHAPTER 3
gallon equals 3.79 liters) and the price of oil (West Texas intermediate crude) was $27.18 per barrel (one barrel equals 42
U.S. gallons). (Sources: U.S Department of Energy.) Over the next 10 years, the price of both oil and gasoline substantially
increased. Many drivers complained that the oil companies were guilty of price gouging; that is, they believed that when the
price of oil increased, the price of gas also increased, but when the price of oil decreased, the decrease in the price of gasoline seemed to lag behind. To determine whether this perception is accurate, we determined the monthly figures for both
commodities. Were oil and gas prices related?
I N T RO D U C T I O N
C
hapter 2 introduced graphical techniques used to summarize and present nominal data. In this chapter, we do the same for interval data. Section 3.1 presents
techniques to describe a set of interval data, Section 3.2 introduces time series
and the method used to present time series data, and Section 3.3 describes the technique we use to describe the relationship between two interval variables. We complete
this chapter with a discussion of how to properly use graphical methods in Section 3.4.
3 . 1 G R A PH I C A L T E C H N I Q U E S
TO
DESCRIBE
A
SET
OF
I N T E RVA L D ATA
In this section, we introduce several graphical methods that are used when the data are
interval. The most important of these graphical methods is the histogram. As you will
see, the histogram not only is a powerful graphical technique used to summarize interval
data but also is used to help explain an important aspect of probability (see Chapter 8).
APPLICATIONS in MARKETING
Pricing
© AP Photo/Paul Sakuma
Traditionally, marketing has been defined in terms of the four P’s: product, price,
promotion, and place. Marketing management is the functional area of business
that focuses on the development of a product, together with its pricing, promotion, and distribution. Decisions are made in these four areas with a view to satisfying the wants and needs of consumers while also satisfying the firm’s objective.
The pricing decision must be addressed both for a new product, and, from
time to time, for an existing product. Anyone buying a product such as a personal
computer has been confronted with a wide variety of prices, accompanied by a correspondingly wide variety of features. From a vendor’s standpoint, establishing the appropriate price and
corresponding set of attributes for a product is complicated and must be done in the context of
the overall marketing plan for the product.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
EXAMPLE
3.1
DATA
Xm03-01
45
Analysis of Long-Distance Telephone Bills
Following deregulation of telephone service, several new companies were created to compete in the business of providing long-distance telephone service. In almost all cases, these
companies competed on price because the service each offered is similar. Pricing a service
or product in the face of stiff competition is very difficult. Factors to be considered
include supply, demand, price elasticity, and the actions of competitors. Long-distance
packages may employ per minute charges, a flat monthly rate, or some combination of the
two. Determining the appropriate rate structure is facilitated by acquiring information
about the behaviors of customers, especially the size of monthly long-distance bills.
As part of a larger study, a long-distance company wanted to acquire information
about the monthly bills of new subscribers in the first month after signing with the
company. The company’s marketing manager conducted a survey of 200 new residential
subscribers and recorded the first month’s bills. These data are listed here. The general
manager planned to present his findings to senior executives. What information can be
extracted from these data?
Long-Distance Telephone Bills
42.19
38.45
29.23
89.35
118.04
110.46
0
72.88
83.05
95.73
103.15
94.52
26.84
93.93
90.26
72.78
101.36
104.8
74.01
56.01
39.21
48.54
93.31
104.88
30.61
22.57
63.7
104.84
6.45
16.47
89.5
13.36
44.16
92.97
99.56
92.62
78.89
87.71
93.57
0
75.71
88.62
99.5
85
0
8.41
70.48
92.88
3.2
115.5
2.42
1.08
76.69
13.62
88.51
55.99
12.24
119.63
23.31
11.05
8.37
7.18
11.07
1.47
26.4
13.26
21.13
95.03
29.04
5.42
77.21
72.47
0
5.64
6.48
6.95
19.6
8.11
9.01
84.77
1.62
91.1
10.88
30.62
100.05
26.97
15.43
29.25
1.88
16.44
109.08
2.45
21.97
17.12
19.7
6.93
10.05
99.03
29.24
15.21
28.77
9.12
118.75
0
13.95
14.34
79.52
2.72
9.63
21.34
104.4
2.88
65.9
20.55
3.43
10.44
21.36
24.42
95.52
6.72
35.32
117.69
106.84
8.4
90.04
3.85
91.56
10.13
5.72
33.69
115.78
0.98
19.45
0
27.21
89.27
14.49
92.17
21
106.59
13.9 114.67
9.22 27.57
109.94 64.78
10.7
45.81
0
56.04
11.27 20.39
72.02 31.77
7.74 94.67
5.04 44.32
33.4
3.69
6.95 19.34
6.48 13.54
11.64 18.89
83.26
1.57
15.42
0
24.49
5.2
89.13
2.8
111.14
5.1
92.64
3.03
53.9
9.16
15.3
75.49
68.69
35
9.12
18.49
84.12
13.68
20.84
100.04
112.94
20.12
53.21
15.3
49.24
9.44
2.67
4.69
41.38
45.77
SOLUTION
Little information can be developed just by casually reading through the 200 observations. The manager can probably see that most of the bills are under $100, but that is
likely to be the extent of the information garnered from browsing through the data. If
he examines the data more carefully, he may discover that the smallest bill is $0 and the
largest is $119.63. He has now developed some information. However, his presentation
to senior executives will be most unimpressive if no other information is produced. For
example, someone is likely to ask how the numbers are distributed between 0 and
119.63. Are there many small bills and few large bills? What is the “typical” bill? Are
the bills somewhat similar or do they vary considerably?
To help answer these questions and others like them, the marketing manager can
construct a frequency distribution from which a histogram can be drawn. In the
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previous section a frequency distribution was created by counting the number of times
each category of the nominal variable occurred. We create a frequency distribution for
interval data by counting the number of observations that fall into each of a series of
intervals, called classes, that cover the complete range of observations. We discuss how
to decide the number of classes and the upper and lower limits of the intervals later. We
have chosen eight classes defined in such a way that each observation falls into one and
only one class. These classes are defined as follows:
Classes
Amounts that are less than or equal to 15
Amounts that are more than 15 but less than or equal to 30
Amounts that are more than 30 but less than or equal to 45
Amounts that are more than 45 but less than or equal to 60
Amounts that are more than 60 but less than or equal to 75
Amounts that are more than 75 but less than or equal to 90
Amounts that are more than 90 but less than or equal to 105
Amounts that are more than 105 but less than or equal to 120
Notice that the intervals do not overlap, so there is no uncertainty about which
interval to assign to any observation. Moreover, because the smallest number is 0 and
the largest is 119.63, every observation will be assigned to a class. Finally, the intervals
are equally wide. Although this is not essential, it makes the task of reading and interpreting the graph easier.
To create the frequency distribution manually, we count the number of observations that fall into each interval. Table 3.1 presents the frequency distribution.
TABLE
3.1
Frequency Distribution of the Long-Distance Bills in Example 3.1
CLASS LIMITS
FREQUENCY
0 to 15*
71
15 to 30
37
30 to 45
13
45 to 60
9
60 to 75
10
75 to 90
18
90 to 105
28
105 to 120
14
Total
200
*Classes contain observations greater than their lower limits (except for the first class) and less than or equal to their upper limits.
Although the frequency distribution provides information about how the numbers
are distributed, the information is more easily understood and imparted by drawing a picture or graph. The graph is called a histogram. A histogram is created by drawing rectangles whose bases are the intervals and whose heights are the frequencies. Figure 3.1
exhibits the histogram that was drawn by hand.
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FIGURE
3.1
47
Histogram for Example 3.1
70
60
Frequency
50
40
30
20
10
0
15
30
45
60
75
90
105
120
Long-distance telephone bills
EXCEL
80
Frequency
60
40
20
0
15
30
45
60
75
90
105
120
Bills
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-01.) In another column, type
the upper limits of the class intervals. Excel calls them bins. (You can put any name in
the first row; we typed “Bills.”)
2. Click Data, Data Analysis, and Histogram. If Data Analysis does not appear in the
menu box, see our Keller’s website, Appendix A1.
3. Specify the Input Range (A1:A201) and the Bin Range (B1:B9). Click Chart
Output. Click Labels if the first row contains names.
4. To remove the gaps, place the cursor over one of the rectangles and click the right
button of the mouse. Click (with the left button) Format Data Series . . . . move the
pointer to Gap Width and use the slider to change the number from 150 to 0.
Except for the first class, Excel counts the number of observations in each class that are
greater than the lower limit and less than or equal to the upper limit.
Note that the numbers along the horizontal axis represent the upper limits of each
class although they appear to be placed in the centers. If you wish, you can replace these
numbers with the actual midpoints by making changes to the frequency distribution in
cells A1:B14 (change 15 to 7.5, 30 to 22.5, . . . , and 120 to 112.5).
You can also convert the histogram to list relative frequencies instead of frequencies.
To do so, change the frequencies to relative frequencies by dividing each frequency by
200; that is, replace 71 by .355, 37 by .185, . . . , and 14 by .07.
If you have difficulty with this technique, turn to the website Appendix A2 or A3,
which provides step-by-step instructions for Excel and provides troubleshooting tips.
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MINITAB
80
70
Frequency
60
50
40
30
20
10
0
0
30
60
90
120
Bills
Note that Minitab counts the number of observations in each class that are strictly less
than their upper limits.
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-01.)
2. Click Graph, Histogram . . . , and Simple.
3. Type or use the Select button to specify the name of the variable in the Graph
Variables box (Bills). Click Data View.
4. Click Data Display and Bars. Minitab will create a histogram using its own choices of
class intervals.
5. To choose your own classes, double-click the horizontal axis. Click Binning.
6. Under Interval Type, choose Cutpoint. Under Interval Definition, choose
Midpoint/Cutpoint positions and type in your choices (0 15 30 45 60 75 90 105
120) to produce the histogram shown here.
INTERPRET
The histogram gives us a clear view of the way the bills are distributed. About half the
monthly bills are small ($0 to $30), a few bills are in the middle range ($30 to $75), and
a relatively large number of long-distance bills are at the high end of the range. It would
appear from this sample of first-month long-distance bills that the company’s customers
are split unevenly between light and heavy users of long-distance telephone service. If
the company assumes that this pattern will continue, it must address a number of pricing issues. For example, customers who incurred large monthly bills may be targets of
competitors who offer flat rates for 15-minute or 30-minute calls. The company needs
to know more about these customers. With the additional information, the marketing
manager may suggest altering the company’s pricing.
Determining the Number of Class Intervals
The number of class intervals we select depends entirely on the number of observations
in the data set. The more observations we have, the larger the number of class intervals
we need to use to draw a useful histogram. Table 3.2 provides guidelines on choosing
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49
the number of classes. In Example 3.1, we had 200 observations. The table tells us to
use 7, 8, 9, or 10 classes.
TABLE
3. 2
Approximate Number of Classes in Histograms
NUMBER OF OBSERVATIONS
NUMBER OF CLASSES
Less than 50
5–7
50–200
7–9
200–500
9–10
500–1,000
10–11
1,000–5,000
11–13
5,000–50,000
13–17
More than 50,000
17–20
An alternative to the guidelines listed in Table 3.2 is to use Sturges’s formula, which
recommends that the number of class intervals be determined by the following:
Number of class intervals = 1 3.3 log (n)
For example, if n = 50 Sturges’s formula becomes
Number of class intervals = 1 3.3 log(50) 1 3.3(1.7) 6.6
which we round to 7.
Class Interval Widths We determine the approximate width of the classes by subtracting the smallest observation from the largest and dividing the difference by the
number of classes. Thus,
Class width =
Largest Observation - Smallest Observation
Number of Classes
In Example 3.1, we calculated
Class width =
119.63 - 0
= 14.95
8
We often round the result to some convenient value. We then define our class limits by
selecting a lower limit for the first class from which all other limits are determined. The
only condition we apply is that the first class interval must contain the smallest observation. In Example 3.1, we rounded the class width to 15 and set the lower limit of the
first class to 0. Thus, the first class is defined as “Amounts that are greater than or equal
to 0 but less than or equal to 15.” (Minitab users should remember that the classes are
defined as the number of observations that are strictly less than their upper limits.)
Table 3.2 and Sturges’s formula are guidelines only. It is more important to choose
classes that are easy to interpret. For example, suppose that we have recorded the marks
on an exam of the 100 students registered in the course where the highest mark is 94
and the lowest is 48. Table 3.2 suggests that we use 7, 8, or 9 classes, and Sturges’s formula computes the approximate number of classes as
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Number of class intervals 1 3.3 log(100) 1 3.3(2) 7.6
which we round to 8. Thus,
Class width =
94 - 48
= 5.75
8
which we would round to 6. We could then produce a histogram whose upper limits of
the class intervals are 50, 56, 62, . . . , 98. Because of the rounding and the way in which
we defined the class limits, the number of classes is 9. However, a histogram that is easier to interpret would be produced using classes whose widths are 5; that is, the upper
limits would be 50, 55, 60, . . . , 95. The number of classes in this case would be 10.
Shapes of Histograms
The purpose of drawing histograms, like that of all other statistical techniques, is to
acquire information. Once we have the information, we frequently need to describe
what we’ve learned to others. We describe the shape of histograms on the basis of the
following characteristics.
Symmetry A histogram is said to be symmetric if, when we draw a vertical line down
the center of the histogram, the two sides are identical in shape and size. Figure 3.2
depicts three symmetric histograms.
Three Symmetric Histograms
Frequency
3.2
Frequency
Frequency
FIGURE
Variable
Variable
Variable
Skewness A skewed histogram is one with a long tail extending to either the right or
the left. The former is called positively skewed, and the latter is called negatively
skewed. Figure 3.3 shows examples of both. Incomes of employees in large firms tend to
be positively skewed because there is a large number of relatively low-paid workers and a
small number of well-paid executives. The time taken by students to write exams is frequently negatively skewed because few students hand in their exams early; most prefer to
reread their papers and hand them in near the end of the scheduled test period.
Positively and Negatively Skewed Histograms
Frequency
3.3
Frequency
FIGURE
Variable
Variable
Number of Modal Classes As we discuss in Chapter 4, a mode is the observation that
occurs with the greatest frequency. A modal class is the class with the largest number of
observations. A unimodal histogram is one with a single peak. The histogram in
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51
Figure 3.4 is unimodal. A bimodal histogram is one with two peaks, not necessarily
equal in height. Bimodal histograms often indicate that two different distributions are
present. (See Example 3.4.) Figure 3.5 depicts bimodal histograms.
3.4
A Unimodal Histogram
Frequency
FIGURE
Variable
Bimodal Histograms
Frequency
3.5
Frequency
FIGURE
Variable
Variable
Bell Shape A special type of symmetric unimodal histogram is one that is bell shaped.
In Chapter 8 we will explain why this type of histogram is important. Figure 3.6 exhibits
a bell-shaped histogram.
3.6
Bell-Shaped Histogram
Frequency
FIGURE
Variable
Now that we know what to look for, let’s examine some examples of histograms and
see what we can discover.
APPLICATIONS in FINANCE
Stock and Bond Valuation
© BanaStock/Jupiter Images
A basic understanding of how financial assets, such as stocks and bonds, are
valued is critical to good financial management. Understanding the basics of
valuation is necessary for capital budgeting and capital structure decisions.
Moreover, understanding the basics of valuing investments such as stocks and
bonds is at the heart of the huge and growing discipline known as investment
management.
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A financial manager must be familiar with the main characteristics of the capital markets
where long-term financial assets such as stocks and bonds trade. A well-functioning capital market provides managers with useful information concerning the appropriate prices and rates of
return that are required for a variety of financial securities with differing levels of risk. Statistical
methods can be used to analyze capital markets and summarize their characteristics, such as the
shape of the distribution of stock or bond returns.
APPLICATIONS in FINANCE
Return on Investment
The return on an investment is calculated by dividing the gain (or loss) by the
value of the investment. For example, a $100 investment that is worth $106
after 1 year has a 6% rate of return. A $100 investment that loses $20 has a
© Vicki Beaver
–20% rate of return. For many investments, including individual stocks and
stock portfolios (combinations of various stocks), the rate of return is a variable.
In other words, the investor does not know in advance what the rate of return will
be. It could be a positive number, in which case the investor makes money—or negative, and the investor loses money.
Investors are torn between two goals. The first is to maximize the rate of return on investment. The second goal is to reduce risk. If we draw a histogram of the returns for a certain investment, the location of the center of the histogram gives us some information about the return one
might expect from that investment. The spread or variation of the histogram provides us with
guidance about the risk. If there is little variation, an investor can be quite confident in predicting
what his or her rate of return will be. If there is a great deal of variation, the return becomes
much less predictable and thus riskier. Minimizing the risk becomes an important goal for
investors and financial analysts.
EXAMPLE
3. 2
DATA
Xm03-02
Comparing Returns on Two Investments
Suppose that you are facing a decision about where to invest that small fortune that
remains after you have deducted the anticipated expenses for the next year from the
earnings from your summer job. A friend has suggested two types of investment, and to
help make the decision you acquire some rates of return from each type. You would like
to know what you can expect by way of the return on your investment, as well as other
types of information, such as whether the rates are spread out over a wide range (making the investment risky) or are grouped tightly together (indicating relatively low risk).
Do the data indicate that it is possible that you can do extremely well with little likelihood of a large loss? Is it likely that you could lose money (negative rate of return)?
The returns for the two types of investments are listed here. Draw histograms for
each set of returns and report on your findings. Which investment would you choose
and why?
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
Returns on Investment A
30.00
2.13
4.30
25.00
12.89
20.24
1.20
2.59
33.00
14.26
15.83
0.63
38.00
6.93
13.24
18.95
9.43
1.21
31.76
11.07
8.47
36.08
21.95
10.33
12.68
13.09
Returns on Investment B
8.55
5.29
7.04
12.11
12.89
63.00
19.27
9.22
17.00
17.30
52.00
13.77
22.42
34.40
49.87
22.92
20.95
43.71
12.83
0.52
61.00
11.96
1.94
28.45
34.75
54.19
44.00
20.23
4.16
10.03
10.51
39.04
24.76
15.28
34.21
52.00
32.17
30.33
30.37
5.61
29.00
26.01
0.46
2.07
29.44
11
25.93
8.29
61.00
20.44
30.31
6.06
14.73
36.13
1.53
17.61
1.2
9.94
33.39
58.67
0.25
5.23
66
24.3
10.01
35.24
40.7
22.18
3.24
25.1
24.24
38.47
13.44
68.00
SOLUTION
We draw the histograms of the returns on the two investments. We’ll use Excel and
Minitab to do the work.
EXCEL
18
16
14
12
10
8
6
4
2
0
Histogram of Returns on Investment B
Frequency
Frequency
Histogram of Returns on Investment A
-30
-15
0
15
30
45
60
75
18
16
14
12
10
8
6
4
2
0
-30
-15
0
15
30
45
60
75
Returns
Returns
MINITAB
Histogram of Returns on Investment B
18
16
16
14
14
12
12
Frequency
Frequency
Histogram of Returns on Investment A
18
10
8
10
8
6
6
4
4
2
2
0
0
-30
-15
0
15
30
Returns
45
60
75
-30
0
30
60
Returns
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INTERPRET
Comparing the two histograms, we can extract the following information:
1. The center of the histogram of the returns of investment A is slightly lower than that
for investment B.
2. The spread of returns for investment A is considerably less than that for investment B.
3. Both histograms are slightly positively skewed.
These findings suggest that investment A is superior. Although the returns for A
are slightly less than those for B, the wider spread for B makes it unappealing
to most investors. Both investments allow for the possibility of a relatively large
return.
The interpretation of the histograms is somewhat subjective. Other viewers may
not concur with our conclusion. In such cases, numerical techniques provide the detail
and precision lacking in most graphs. We will redo this example in Chapter 4 to illustrate how numerical techniques compare to graphical ones.
EXAMPLE
3.3
DATA
Xm03-03*
Business Statistics Marks
A student enrolled in a business program is attending the first class of the required statistics course. The student is somewhat apprehensive because he believes the myth that
the course is difficult. To alleviate his anxiety, the student asks the professor about last
year’s marks. The professor obliges and provides a list of the final marks, which is composed of term work plus the final exam. Draw a histogram and describe the result, based
on the following marks:
65
71
66
79
65
82
80
86
67
64
62
74
67
72
68
81
53
70
76
73
73
85
83
80
67
78
68
67
62
83
72
85
72
77
64
77
89
87
78
79
59
63
84
74
74
59
66
71
68
72
75
74
77
69
60
92
69
69
73
65
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55
SOLUTION
EXCEL
30
Frequency
25
20
15
10
5
0
50
60
70
80
Marks
90
100
MINITAB
25
Frequency
20
15
10
5
0
50
60
70
80
90
100
Marks
INTERPRET
The histogram is unimodal and approximately symmetric. There are no marks below
50, with the great majority of marks between 60 and 90. The modal class is 70 to 80,
and the center of the distribution is approximately 75.
EXAMPLE
3.4
DATA
Xm03-04*
Mathematical Statistics Marks
Suppose the student in Example 3.3 obtained a list of last year’s marks in a mathematical statistics course. This course emphasizes derivations and proofs of theorems. Use
the accompanying data to draw a histogram and compare it to the one produced in
Example 3.3. What does this histogram tell you?
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77
74
75
75
67
72
81
76
79
73
59
83
77
74
78
67
82
55
73
92
75
75
52
72
78
83
81
73
72
71
53
75
76
59
82
82
70
71
75
74
53
49
56
61
61
54
44
54
60
50
52
47
46
50
51
44
52
53
56
53
SOLUTION
EXCEL
30
Frequency
25
20
15
10
5
0
50
60
70
80
Marks
90
100
MINITAB
25
Frequency
20
15
10
5
0
40
50
60
70
80
90
100
Marks
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57
INTERPRET
The histogram is bimodal. The larger modal class is composed of the marks in the 70s.
The smaller modal class includes the marks that are in the 50s. There appear to be few
marks in the 60s. This histogram suggests that there are two groups of students.
Because of the emphasis on mathematics in the course, one may conclude that those
who performed poorly in the course are weaker mathematically than those who performed well. The histograms in this example and in Example 3.3 suggest that the
courses are quite different from one another and have a completely different distribution of marks.
Stem-and-Leaf Display
One of the drawbacks of the histogram is that we lose potentially useful information by
classifying the observations. In Example 3.1, we learned that there are 71 observations
that fall between 0 and 15. By classifying the observations we did acquire useful information. However, the histogram focuses our attention on the frequency of each class
and by doing so sacrifices whatever information was contained in the actual observations. A statistician named John Tukey introduced the stem-and-leaf display, which is
a method that to some extent overcomes this loss.
The first step in developing a stem-and-leaf display is to split each observation into
two parts, a stem and a leaf. There are several different ways of doing this. For example,
the number 12.3 can be split so that the stem is 12 and the leaf is 3. In this definition the
stem consists of the digits to the left of the decimal and the leaf is the digit to the right of
the decimal. Another method can define the stem as 1 and the leaf as 2 (ignoring the 3).
In this definition the stem is the number of tens and the leaf is the number of ones. We’ll
use this definition to create a stem-and-leaf display for Example 3.1.
The first observation is 42.19. Thus, the stem is 4 and the leaf is 2. The second observation is 38.45, which has a stem of 3 and a leaf of 8. We continue converting each number in this way. The stem-and-leaf display consists of listing the stems 0, 1, 2, . . . , 11.
After each stem, we list that stem’s leaves, usually in ascending order. Figure 3.7 depicts
the manually created stem-and-leaf display.
FIGURE
Stem
0
1
2
3
4
5
6
7
8
9
10
11
3.7
Stem-and-Leaf Display for Example 3.1
Leaf
000000000111112222223333345555556666666778888999999
000001111233333334455555667889999
0000111112344666778999
001335589
124445589
33566
3458
022224556789
334457889999
00112222233344555999
001344446699
0124557889
As you can see the stem-and-leaf display is similar to a histogram turned on its side. The
length of each line represents the frequency in the class interval defined by the stems.
The advantage of the stem-and-leaf display over the histogram is that we can see the
actual observations.
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CHAPTER 3
EXCEL
A
B
C
D
E
F
G
1 Stem & Leaf Display
2
3
Stems
Leaves
4
0 ->0000000001111122222233333455555566666667788889999999
5
1 ->000001111233333334455555667889999
6
2 ->00001111123446667789999
3 ->001335589
7
8
4 ->12445589
9
5 ->33566
10
6 ->3458
11
7 ->022224556789
12
8 ->334457889999
13
9 ->00112222233344555999
10 ->001344446699
14
15
11 ->0124557889
16
17
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-01.)
2. Click Add-ins, Data Analysis Plus, and Stem-and-Leaf Display.
3. Specify the Input Range (A1:A201). Click one of the values of Increment (the increment is the difference between stems) (10).
MINITAB
Stem-and-Leaf Display: Bills
Stem-and-leaf of Bills N = 200
Leaf Unit = 1.0
52
85
(23)
92
83
75
70
66
54
42
22
10
0
1
2
3
4
5
6
7
8
9
10
11
0000000001111122222233333455555566666667788889999999
000001111233333334455555667889999
00001111123446667789999
001335589
12445589
33566
3458
022224556789
334457889999
00112222233344555999
001344446699
0124557889
The numbers in the left column are called depths. Each depth counts the number of
observations that are on its line or beyond. For example, the second depth is 85, which
means that there are 85 observations that are less than 20. The third depth is displayed in
parentheses, which indicates that the third interval contains the observation that falls in the
middle of all the observations, a statistic we call the median (to be presented in Chapter 4).
For this interval, the depth tells us the frequency of the interval; that is, 23 observations are
greater than or equal to 20 but less than 30. The fourth depth is 92, which tells us that 92
observations are greater than or equal to 30. Notice that for classes below the median, the
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
59
depth reports the number of observations that are less than the upper limit of that class. For
classes that are above the median, the depth reports the number of observations that are
greater than or equal to the lower limit of that class.
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-01.)
2. Click Graph and Stem-and-Leaf. . . .
3. Type or use the Select button to specify the variable in the Variables box (Bills). Type
the increment in the Increment box (10).
Ogive
The frequency distribution lists the number of observations that fall into each class
interval. We can also create a relative frequency distribution by dividing the frequencies by the number of observations. Table 3.3 displays the relative frequency distribution for Example 3.1.
TABLE
3. 3
Relative Frequency Distribution for Example 3.1
CLASS LIMITS
RELATIVE FREQUENCY
0 to 15
71/200 .355
15 to 30
37/200 .185
30 to 45
13/200 .065
45 to 60
9/200 .045
60 to 75
10/200 .050
75 to 90
18/200 .090
90 to 105
28/200 .140
105 to 120
14/200 .070
Total
200/200 1.0
As you can see, the relative frequency distribution highlights the proportion of the
observations that fall into each class. In some situations, we may wish to highlight the
proportion of observations that lie below each of the class limits. In such cases, we create the cumulative relative frequency distribution. Table 3.4 displays this type of distribution for Example 3.1.
From Table 3.4, you can see that, for example, 54% of the bills were less than or
equal to $30 and that 79% of the bills were less than or equal to $90.
Another way of presenting this information is the ogive, which is a graphical representation of the cumulative relative frequencies. Figure 3.8 is the manually drawn
ogive for Example 3.1.
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CHAPTER 3
TABLE
3. 4
Cumulative Relative Frequency Distribution for Example 3.1
CLASS LIMITS
RELATIVE FREQUENCY
CUMULATIVE
RELATIVE FREQUENCY
0 to 15
71/200 .355
71/200 .355
15 to 30
37/200 .185
108/200 .540
30 to 45
13/200 .065
121/200 .605
9/200 .045
130/200 .650
45 to 60
60 to 75
10/200 .05
140/200 .700
75 to 90
18/200 .09
158/200 .790
90 to 105
28/200 .14
186/200 .930
105 to 120
14/200 .07
200/200 1.00
FIGURE
3.8
Ogive for Example 3.1
1.000
Cumulative relative frequency
1.00
.930
.90
.790
.80
.650
.70
.700
.605
.60
.540
.50
.40
.355
.30
.20
.10
0
15
30
45
60
75
90 105
Long-distance telephone bills
120
EXCEL
Ogive
Percentages
100
80
60
40
20
0
0
15
30
45
60
75
90
105 120
Bills
INSTRUCTIONS
Follow instructions to create a histogram. Make the first bin’s upper limit a number that
is slightly smaller than the smallest number in the data set. Move the cursor to Chart
Output and click. Do the same for Cumulative Percentage. Remove the “More”
category. Click on any of the rectangles and click Delete. Change the Scale, if necessary. (Right-click the vertical or horizontal axis, click Format Axis . . . , and change the
Maximum value of Y equal to 1.0.)
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
61
MINITAB
Minitab does not draw ogives.
We can use the ogive to estimate the cumulative relative frequencies of other values. For example, we estimate that about 62% of the bills lie below $50 and that about
48% lie below $25. (See Figure 3.9.)
3.9
FIGURE
Ogive with Estimated Relative Frequencies for Example 3.1
1.000
Cumulative relative frequency
1.00
.930
.90
.790
.80
.650
.70
.60
.540
.700
.605
.50
.40
.355
.30
.20
.10
0
15
30
45
60
75
90
105
Long-distance telephone bills
120
Here is a summary of this section’s techniques.
Factors That Identify When to Use a Histogram, Ogive, or
Stem-and-Leaf Display
1. Objective: Describe a single set of data
2. Data type: Interval
EXERCISES
3.1 How many classes should a histogram contain if the
number of observations is 250?
3.2 Determine the number of classes of a histogram for
700 observations.
3.3 A data set consists of 125 observations that range
between 37 and 188.
a. What is an appropriate number of classes to have
in the histogram?
b. What class intervals would you suggest?
3.4 A statistics practitioner would like to draw a histogram
of 62 observations that range from 5.2 to 6.1.
a. What is an appropriate number of class intervals?
b. Define the upper limits of the classes you would use.
3.5 Xr03-05 The number of items rejected daily by a
manufacturer because of defects was recorded for
the past 30 days. The results are as follows.
4
8
5
9 13
15 17
3
9
7
19
19
5
6
14
8
4
13
12
10
18
15
8
7
5
7
22 16
3
9
a. Construct a histogram.
b. Construct an ogive.
c. Describe the shape of the histogram.
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CHAPTER 3
3.6 Xr03-06 The final exam in a third-year organizational
3
2
8
5
3
0
6
4
4
5
behavior course requires students to write several
essay-style answers. The numbers of pages for a
sample of 25 exams were recorded. These data are
shown here.
5 8
7 12
9
9
3
6
12
3
8
8
5 7 3 8
7 10 9 12
9 5
7 3
2
a. Draw a histogram.
b. Draw an ogive.
c. Describe what you’ve learned from the answers
to parts (a) and (b).
3.7 Xr03-07 A large investment firm on Wall Street wants
to review the distribution of ages of its stockbrokers.
The firm believes that this information can be useful
in developing plans to recruit new brokers. The ages
of a sample of 40 brokers are shown here.
46
53
36
41
a.
b.
c.
d.
28
40
50
37
51
50
46
26
34
33
33
39
29
36
61
35
40
41
48
39
38
25
32
46
33
38
28
26
41
37
30
31
52
41
49
35
9
17
10
3 25
19 10
15 10
17
15
8
3.9 Xr03-09 The amount of time (in seconds) needed to
complete a critical task on an assembly line was measured for a sample of 50 assemblies. These data are
as follows:
30.3
31.9
34.4
32.8
31.1
32.2
30.7
36.8
30.5
30.6
34.5
33.1
30.1
31.0
30.7
30.9
30.7
30.2
30.6
37.9
31.1
31.1
34.6
30.2
33.1
32.1
30.6
31.5
30.2
30.3
30.9
30.0
31.6
30.2
34.4
34.2
30.2
30.1
31.4
34.1
5
16
11
8
8
8
5
11
14
10
3.10 Xr03-10 A survey of individuals in a mall asked 60
people how many stores they will enter during this
visit to the mall. The responses are listed here.
9
2
1
7
8
4
8
1
6
3
15
26
15
19
7
6
24
20
9
18
14
6
6
16
10
6
5
24
25
22
7
15
4
9
5
21
28
5
10
12
8
23
7
9
8
10
9
13
24
17
3.12 Xr03-12 To help determine the need for more golf
courses, a survey was undertaken. A sample of 75
self-declared golfers was asked how many rounds
of golf they played last year. These data are as
follows:
18
15
25
18
29
28
23
20
13
26
29
24
3
13
28
33.7
32.7
32.4
32.8
31.0
30.7
33.4
35.7
30.7
30.4
a. Draw a stem-and-leaf display.
b. Draw a histogram.
c. Describe the histogram.
3
2
5
1
4
4
3
9
4
4
number of games they attended last year. The results
are listed here. Use an appropriate graphical technique to present these data and describe what you
have learned.
sample of 30 telemarketers are listed here. Draw
a histogram of these data and describe it.
21 4
16 3
19 7
3
6
4
1
1
4
5
6
0
1
3.11 Xr03-11 A survey asked 50 baseball fans to report the
3.8 Xr03-08 The numbers of weekly sales calls by a
8 6 12
5 8 18
20 17 14
4
3
6
3
1
5
2
1
1
5
a. Draw a histogram.
b. Draw an ogive.
c. Describe your findings.
Draw a stem-and-leaf display.
Draw a histogram.
Draw an ogive.
Describe what you have learned.
14
9
5
2
4
7
2
4
2
2
3
4
5
a.
b.
c.
d.
26
18
30
24
23
9
20
30
26
17
18
30
28
18
19
16
15
35
21
15
17
24
26
22
32
38
20
5
10
10
35
18
14
25
19
28
28
12
30
36
31
13
14
14
42
30
29
20
18
27
25
36
31
29
24
36
23
24
16
22
Draw a histogram.
Draw a stem-and-leaf display.
Draw an ogive.
Describe what you have learned.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
3.17 Xr03-17 The marks of 320 students on an econom-
The following exercises require a computer and statistical
software.
3.13
ics midterm test were recorded. Use a graphical
technique to summarize these data. What does
the graph tell you?
The annual incomes for a sample of 200 firstyear accountants were recorded. Summarize these
data using a graphical method. Describe your results.
Xr03-13
3.18 Xr03-18 The lengths (in inches) of 150 newborn
babies were recorded. Use whichever graphical
technique you judge suitable to describe these data.
What have you learned from the graph?
3.14 Xr03-14 The real estate board in a suburb of
Los Angeles wanted to investigate the distribution
of the prices (in $ thousands) of homes sold during
the past year.
a Draw a histogram.
b. Draw an ogive.
c. Draw a stem-and-leaf display (if your software
allows it).
d. Describe what you have learned.
3.19 Xr03-19 The number of copies made by an office
copier was recorded for each of the past 75 days.
Graph the data using a suitable technique.
Describe what the graph tells you.
3.20 Xr03-20 Each of a sample of 240 tomatoes grown
with a new type of fertilizer was weighed (in
ounces) and recorded. Draw a histogram and
describe your findings.
3.15 Xr03-15 The number of customers entering a bank in
the first hour of operation for each of the last 200
days was recorded. Use a graphical technique to
extract information. Describe your findings.
3.16
63
3.21 Xr03-21 The volume of water used by each of a
sample of 350 households was measured (in gallons) and recorded. Use a suitable graphical statistical method to summarize the data. What does
the graph tell you?
The lengths of time (in minutes) to serve 420
customers at a local restaurant were recorded.
a. How many bins should a histogram of these data
contain?
b. Draw a histogram using the number of bins specified in part (a).
c. Is the histogram symmetric or skewed?
d. Is the histogram bell shaped?
Xr03-16
3.22 Xr03-22 The number of books shipped out daily by
Amazon.com was recorded for 100 days. Draw a
histogram and describe your findings.
APPLICATIONS in BANKING
© Comstock Images/Jupiterimages
Credit Scorecards
Credit scorecards are used by banks and financial institutions to determine whether
applicants will receive loans. The scorecard is the product of a statistical technique
that converts questions about income, residence, and other variables into a score.
The higher the score, the higher the probability that the applicant will repay. The
scorecard is a formula produced by a statistical technique called logistic regression,
which is available as an appendix on the Keller’s website. For example, a scorecard may
score age categories in the following way:
Less than 25
25 to 39
40 to 55
Over 55
20 points
24
30
38
Other variables would be scored similarly. The sum for all variables would be the applicant’s score. A cutoff score would be used to predict those who will repay and those who will
default. Because no scorecard is perfect, it is possible to make two types of error: granting
credit to those who will default and not lending money to those who would have repaid.
(Continued)
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EXERCISES
3.23 Xr03-23 A small bank that had not yet used a scorecard wanted to determine
whether a scorecard would be advantageous. The bank manager took a random
sample of 300 loans that were granted and scored each on a scorecard borrowed
from a similar bank. This scorecard is based on the responses supplied by the
applicants to questions such as age, marital status, and household income. The
cutoff is 650, which means that those scoring below are predicted to default and
those scoring above are predicted to repay. Two hundred twenty of the loans were
repaid, the rest were not. The scores of those who repaid and the scores of those
who defaulted were recorded.
a. Use a graphical technique to present the scores of those who repaid.
b. Use a graphical technique to present the scores of those who defaulted.
c. What have you learned about the scorecard?
3.24 Xr03-24 Refer to Exercise 3.23. The bank decided to try another scorecard, this
one based not on the responses of the applicants but on credit bureau reports,
which list problems such as late payments and previous defaults. The scores using
the new scorecard of those who repaid and the scores of those who did not repay
were recorded. The cutoff score is 650.
a.
b.
c.
d.
Use a graphical technique to present the scores of those who repaid.
Use a graphical technique to present the scores of those who defaulted.
What have you learned about the scorecard?
Compare the results of this exercise with those of Exercise 3.23. Which scorecard appears to be better?
GENER AL SO CIAL SURVEY EXERCISES
3.25 The GSS asked respondents to specify their highest
year of school completed (EDUC).
a. Is this type of data interval, ordinal, or nominal?
b. GSS2008* Use a graphical technique to present
these data for the 2008 survey.
c. Briefly describe your results.
3.26 GSS2008* Graphically display the results of the GSS
3.27 GSS2008* Employ a graphical technique to present
the ages (AGE) of the respondents in the 2008 survey. Describe your results.
3.28 GSS2008* The survey in 2008 asked “If working, full-
or part-time, how many hours did you work last
week at all jobs (HRS)?” Summarize these data with
a graphical technique.
2008 question, On average days how many hours do
you spend watching television (TVHOURS)?
Briefly describe what you have discovered.
3 . 2 D E S C R I B I N G T I M E -S E R I E S D ATA
Besides classifying data by type, we can also classify them according to whether the
observations are measured at the same time or whether they represent measurements at
successive points in time. The former are called cross-sectional data, and the latter
time-series data.
The techniques described in Section 3.1 are applied to cross-sectional data. All the
data for Example 3.1 were probably determined within the same day. We can probably
say the same thing for Examples 3.2 to 3.4.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
65
To give another example, consider a real estate consultant who feels that the selling
price of a house is a function of its size, age, and lot size. To estimate the specific form of the
function, she samples, say, 100 homes recently sold and records the price, size, age, and lot
size for each home. These data are cross-sectional: They all are observations at the same
point in time. The real estate consultant is also working on a separate project to forecast the
monthly housing starts in the northeastern United States over the next year. To do so, she
collects the monthly housing starts in this region for each of the past 5 years. These 60 values (housing starts) represent time-series data because they are observations taken over time.
Note that the original data may be interval or nominal. All the illustrations above
deal with interval data. A time series can also list the frequencies and relative frequencies
of a nominal variable over a number of time periods. For example, a brand-preference
survey asks consumers to identify their favorite brand. These data are nominal. If we
repeat the survey once a month for several years, the proportion of consumers who prefer a certain company’s product each month would constitute a time series.
Line Chart
Time-series data are often graphically depicted on a line chart, which is a plot of the
variable over time. It is created by plotting the value of the variable on the vertical axis
and the time periods on the horizontal axis.
The chapter-opening example addresses the issue of the relationship between the
price of gasoline and the price of oil. We will introduce the technique we need to
answer the question in Section 3.3. Another question arises: Is the recent price of gasoline high compared to the past prices?
EXA MPLE 3.5
DATA
Xm03-05
Price of Gasoline
We recorded the monthly average retail price of gasoline (in cents per gallon) since
January 1976. Some of these data are displayed below. Draw a line chart to describe
these data and briefly describe the results.
Year
Month
1976
1
Price per gallon
60.5
1976
2
60.0
1976
3
59.4
1976
4
59.2
1976
5
60.0
1976
6
61.6
1976
7
62.3
1976
8
62.8
1976
9
63.0
1976
10
62.9
1976
11
62.9
1976
12
62.6
2009
1
178.7
2009
2
192.8
2009
3
194.9
2009
4
205.6
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CHAPTER 3
2009
5
226.5
2009
6
263.1
2009
7
254.3
2009
8
262.7
2009
9
257.4
2009
10
256.1
2009
11
266.0
2009
12
260.0
SOLUTION
Here are the line charts produced manually, and by Excel and Minitab.
FIGURE
3.10
Line Chart for Example 3.5
Price per gallon
450.0
400.0
350.0
300.0
250.0
200.0
150.0
100.0
50.0
0.0
1
17
33
49
65
81
97 113 129 145 161 177 193 209 225 241 257 273 289 305 321 337 353 369 385 401
EXCEL
Average price of gasoline
450
Line Chart
400
350
300
250
200
150
100
50
0
1
25 49 73 97 121 145 169 193 217 241 265 289 313 337 361 385
Month
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
67
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-05.)
2. Highlight the column of data. Click Insert, Line, and the first 2-D Line. Click
Chart Tools and Layout to make whatever changes you wish.
You can draw two or more line charts (for two or more variables) by highlighting all
columns of data you wish to graph.
MINITAB
Time Series Plot of Price per gallon
450
400
Price per gallon
350
300
250
200
150
100
50
1
41
82
123
164
205
246
287
328
369
Index
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-05.)
2. Click Graph and Time Series Plot . . . . Click Simple.
3. In the series box type or use the Select button to specify the variable (Price). Click
Time/Scale.
4. Click the Time tab, and under Time Scale click Index.
INTERPRET
The price of gasoline rose from about $.60 to more than a dollar in the late 1970s
(months 1 to 49), fluctuated between $.90 and $1.50 until 2000 (months 49 to 289),
then generally rose with large fluctuations (months 289 to 380), then declined sharply
before rallying in the last 10 months.
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CHAPTER 3
APPLICATIONS in ECONOMICS
Measuring Inflation: Consumer Price Index*
Inflation is the increase in the prices for goods and services. In most countries, inflation is measured using the Consumer Price Index (CPI). The Consumer Price Index works with a basket of
some 300 goods and services in the United States (and a similar number in other countries),
including such diverse items as food, housing, clothing, transportation, health, and recreation.
The basket is defined for the “typical” or “average” middle-income family, and the set of items
and their weights are revised periodically (every 10 years in the United States and every
7 years in Canada).
Prices for each item in this basket are computed on a monthly basis and the CPI is computed
from these prices. Here is how it works. We start by setting a period of time as the base. In the
United States the base is the years 1982–1984. Suppose that the basket of goods and services
cost $1,000 during this period. Thus, the base is $1,000, and the CPI is set at 100. Suppose that in
the next month (January 1985) the price increases to $1,010. The CPI for January 1985 is calculated in the following way:
CPI(January 1985) =
1,010
* 100 = 101
1,000
If the price increases to $1,050 in the next month, the CPI is
CPI(February 1985) =
1,050
* 100 = 105
1,000
The CPI, despite never really being intended to serve as the official measure of inflation, has
come to be interpreted in this way by the general public. Pension-plan payments, old-age Social
Security, and some labor contracts are automatically linked to the CPI and automatically indexed
(so it is claimed) to the level of inflation. Despite its flaws, the CPI is used in numerous applications.
One application involves adjusting prices by removing the effect of inflation, making it possible to
track the “real” changes in a time series of prices.
In Example 3.5, the figures shown are the actual prices measured in what are called current
dollars. To remove the effect of inflation, we divide the monthly prices by the CPI for that month
and multiply by 100. These prices are then measured in constant 1982–1984 dollars. This makes it
easier to see what has happened to the prices of the goods and services of interest.
We created two data sets to help you calculate prices in constant 1982–1984 dollars. File
Ch03:\\CPI-Annual and Ch03:\\CPI-Monthly list the values of the CPI where 1982–1984 is set at
100 for annual values and monthly values, respectively.
*Keller’s website Appendix Index Numbers, located at www.cengage.com/bstatistics/keller, describes
index numbers and how they are calculated.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
EXA MPLE 3.6
DATA
Xm03-06
69
Price of Gasoline in 1982–1984 Constant Dollars
Remove the effect of inflation in Example 3.5 to determine whether gasoline prices are
higher than they have been in the past.
SOLUTION
Here are the 1976 and 2009 average monthly prices of gasoline, the CPI, and the
adjusted prices.
The adjusted figures for all months were used in the line chart produced by Excel.
Minitab’s chart is similar.
Year
Month
Price per gallon
CPI
Adjusted price
1976
1976
1976
1976
1976
1976
1976
1976
1976
1976
1976
1976
1
2
3
4
5
6
7
8
9
10
11
12
60.5
60.0
59.4
59.2
60.0
61.6
62.3
62.8
63.0
62.9
62.9
62.6
55.8
55.9
56.0
56.1
56.4
56.7
57.0
57.3
57.6
57.9
58.1
58.4
108.4
107.3
106.1
105.5
106.4
108.6
109.3
109.6
109.4
108.6
108.3
107.2
2009
2009
2009
2009
2009
2009
2009
2009
2009
2009
2009
2009
1
2
3
4
5
6
7
8
9
10
11
12
178.7
192.8
194.9
205.6
226.5
263.1
254.3
262.7
257.4
256.1
266.0
260.0
212.17
213.01
212.71
212.67
212.88
214.46
214.47
215.43
215.79
216.39
217.25
217.54
84.2
90.5
91.6
96.7
106.4
122.7
118.6
121.9
119.3
118.4
122.4
119.5
Ajusted price of gasoline
EXCEL
200
180
160
140
120
100
80
60
40
20
0
1
25 49 73 97 121 145 169 193 217 241 265 289 313 337 361 385
Month
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CHAPTER 3
INTERPRET
Using constant 1982–1984 dollars, we can see that the average price of a gallon of gasoline hit its peak in the middle of 2008 (month 390). From there it dropped rapidly and
in late 2009 was about equal to the adjusted price in 1976.
There are two more factors to consider in judging whether the price of gasoline is
high. The first is distance traveled and the second is fuel consumption. Exercise 3.41
deals with this issue.
EXERCISES
3.29 Xr03-29 The fees television broadcasters pay to cover
the summer Olympic Games has become the largest
source of revenue for the host country. Below we list
the year, city, and revenue in millions of U.S. dollars
paid by television broadcasters around the world.
Draw a chart to describe these prices paid by the
networks.
Year
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
2008
City
Rome
Tokyo
Mexico City
Munich
Montreal
Moscow
Los Angeles
Seoul
Barcelona
Atlanta
Sydney
Athens
Beijing
Broadcast Revenue
1.2
1.6
9.8
17.8
34.9
88.0
266.9
402.6
636.1
898.3
1331.6
1494.0
1737.0
1971
1972
1973
1974
1975
1976
1977
1978
11.8
12.3
16.5
26.3
37.7
43.8
46.1
50.5
61.4
61.5
62.9
63.4
63.2
63.5
61.0
57.9
58.5
58.8
health care than any other country. To gauge how
fast costs are increasing, the following table was produced, listing the total health-care expenditures in
the United States annually for 1981 to 2006 (costs
are in $billions).
a. Graphically present these data.
b. Use the data in CPI-Annual to remove the effect
of inflation. Graph the results and describe your
findings.
United States Army from 1971 to 2007 are listed
here. Draw a line chart, and describe what the chart
tells you.
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
3.31 Xr03-31 The United States spends more money on
3.30 Xr03-30 The number of females enlisted in the
Females
enlisted
55.2
61.7
65.3
64.1
66.5
67.1
68.4
69.7
71.6
72.0
74.3
Source: Statistical Abstract of the United States, 2009, Table 494.
Source: Bloomberg News.
Year
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
Females
enlisted
1990
71.2
1991
67.8
1992
61.7
1993
60.2
1994
59.0
1995
57.3
1996
59.0
1997
62.4
(Continued)
Year
Health
expenditures
Year
Health
expenditures
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
294
331
365
402
440
472
513
574
639
714
782
849
913
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
962
1017
1069
1125
1191
1265
1353
1470
1603
1732
1852
1973
2106
Source: Statistical Abstract of the United States, 2009, Table 124.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
3.32 Xr03-32 The number of earned degrees (thousands)
for males and females is listed below for the years
1987 to 2006. Graph both sets of data. What do the
graphs tell you?
Year
Female
Male
1987
941
882
1988
954
881
1989
986
887
1990
1035
905
1991
1097
928
1992
1147
961
1993
1182
985
1994
1211
995
1995
1223
995
1996
1255
993
1997
1290
998
1998
1304
994
1999
1330
993
2000
1369
1016
2001
1391
1025
2002
1441
1053
2003
1517
1104
2004
1603
1152
2005
1666
1185
2006
1725
1211
Source: U.S. National Center for Education Statistics, Statistical Abstract of the
United States, 2009, Table 288.
3.33 Xr03-33 The number of property crimes (burglary,
larceny, theft, car theft) (in thousands) for the years
1992 to 2006 are listed next. Draw a line chart and
interpret the results.
Year
Crimes
Year
Crimes
1992
1993
1994
1995
1996
1997
1998
1999
12506
12219
12132
12064
11805
11558
10952
10208
2000
2001
2002
2003
2004
2005
2006
10183
10437
10455
10443
10319
10175
9984
Source: U.S. Federal Bureau of Investigation Statistical Abstract of the United
States, 2009, Table 295.
3.34 Xr03-34 Refer to Exercise 3.33. Another way of mea-
suring the number of property crimes is to calculate
71
the number of crimes per 100,000 of population.
This allows us to remove the effect of the increasing
population. Graph these data and interpret your
findings.
Year
Crimes
Year
Crimes
1992
1993
1994
1995
1996
1997
1998
1999
4868
4695
4605
4526
4378
4235
3966
3655
2000
2001
2002
2003
2004
2005
2006
3606
3658
3628
3589
3515
3434
3337
3.35 Xr03-35 The gross national product (GNP) is the
sum total of the economic output of a the citizens
(nationals) of a country. It is an important measure
of the wealth of a country. The following table lists
the year and the GNP in billions of current dollars
for the United States.
a. Graph the GNP. What have you learned?
b. Use the data in CPI-Annual to compute the
per capita GNP in constant 1982–1984
dollars. Graph the results and describe your
findings.
Year
GNP
Year
GNP
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
2822
3160
3290
3572
3967
4244
4478
4754
5124
5508
5835
6022
6371
6699
7109
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
7444
7870
8356
8811
9381
9989
10338
10691
11211
11959
12736
13471
14193
14583
Source: U.S. Bureau of Economic Activity.
3.36 Xr03-36 The average daily U.S. oil consumption and
production (thousands of barrels) is shown for the
years 1973 to 2007. Use a graphical technique to
describe these figures. What does the graph tell
you?
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CHAPTER 3
Year
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
Consumption
17,318
16,655
16,323
17,460
18,443
18,857
18,527
17,060
16,061
15,301
Production
9,209
8,776
8,376
8,132
8,245
8,706
8,551
8,597
8,572
8,649
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
Consumption
15,228
15,722
15,726
16,277
16,666
17,284
17,327
16,988
16,710
17,031
Production
8,689
8,879
8,972
8,683
8,349
8,140
7,615
7,356
7,418
7,172
Year
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
Consumption
17,328
17,721
17,730
18,308
18,618
18,913
19,515
19,699
19,647
19,758
Production
6,847
6,662
6,561
6,465
6,452
6,253
5,882
5,822
5,801
5,746
Year
2003
2004
2005
2006
2007
Consumption
20,034
20,731
20,799
20,800
20,680
5,682
5,419
5,179
5,102
5,065
Production
Source: U.S. Department of Energy: Monthly Energy Review.
3.37 Xr03-37 Has housing been a hedge against infla-
tion in the last 20 years? To answer this question,
we produced the following table, which lists the
average selling price of one-family homes in all of
the United States, the Northeast, Midwest,
South, and West for the years 1988 to 2007, as
well as the annual CPI. For the entire country
and for each area, use a graphical technique to
determine whether housing prices stayed ahead
of inflation.
Year
All
Northeast
Midwest
South
West
CPI
1988
89,300
143,000
68,400
82,200
124,900
118.3
1989
94,600
147,700
73,100
85,600
138,400
124.0
1990
97,300
146,200
76,700
86,300
141,200
130.7
1991
102,700
149,300
81,000
89,800
147,400
136.2
1992
105,500
149,000
84,600
92,900
143,300
140.3
1993
109,100
149,300
87,600
95,800
144,400
144.5
1994
113,500
149,300
90,900
97,200
151,900
148.2
1995
117,000
146,500
96,500
99,200
153,600
152.4
1996
122,600
147,800
102,800
105,000
160,200
156.9
1997
129,000
152,400
108,900
111,300
169,000
160.5
1998
136,000
157,100
116,300
118,000
179,500
163.0
1999
141,200
160,700
121,600
122,100
189,400
166.6
2000
147,300
161,200
125,600
130,300
199,200
172.2
2001
156,600
169,400
132,300
139,600
211,700
177.1
2002
167,600
190,100
138,300
149,700
234,300
179.9
2003
180,200
220,300
143,700
159,700
254,700
184.0
2004
195,200
254,400
151,500
171,800
289,100
188.9
2005
219,000
281,600
168,300
181,100
340,300
195.3
2006
221,900
280,300
164,800
183,700
350,500
201.6
2007
217,900
288,100
161,400
178,800
342,500
207.3
Source: Statistical Abstract of the United States, 2009, Table 935.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
73
b. Calculate the difference between receipts and
outlays. If the difference is positive the result is a
surplus; if the difference is negative the result is a
deficit. Graph the surplus/deficit variable and
describe the results.
3.38 Xr03-38 How has the size of government changed?
To help answer this question, we recorded the U.S.
federal budget receipts and outlays (billions of current dollars) for the years 1980 to 2007.
a. Use a graphical technique to describe the
receipts and outlays of the annual U.S. federal
government budgets since 1980.
Year
1980
1981
1982
1983
1984
1985
1986
Receipts
517.1
599.3
617.8
600.6
666.5
734.1
769.2
Outlays
590.9
678.2
745.7
808.4
851.9
946.4
Year
1990
1991
1992
1993
1994
1987
1988
1989
854.4
909.3
991.2
990.4
1,004.1
1,064.5
1,143.6
1995
1996
1997
1998
1999
Receipts
1,032.0
1,055.0
1,091.3
1,154.4
1,258.6
1,351.8
1,453.1
1,579.3
1,721.8
1,827.5
Outlays
1,253.2
1,324.4
1,381.7
1,409.5
1,461.9
1,515.8
1,560.5
1,601.3
1,652.6
1,701.9
2000
2001
2002
2003
2004
2005
2006
2007
Receipts
2,025.2
1,991.2
1,853.2
1,782.3
1,880.1
2,153.9
2,407.3
2,568.2
Outlays
1,789.1
1,863.9
2,011.0
2,159.9
2,293.0
2,472.2
2,655.4
2,730.2
Year
Source: Statistical Abstract of the United States, 2009, Table 451.
3.39 Refer to Exercise 3.38. Another way of judging the
3.41 Xr03-41 Refer to Example 3.5. The following table
size of budget surplus/deficits is to calculate the
deficit as a percentage of GNP. Use the data in
Exercises 3.35 and 3.38 to calculate this variable and
use a graphical technique to display the results.
lists the average gasoline consumption in miles per
gallon (MPG) and the average distance (thousands
of miles) driven by cars in each of the years 1980 to
2006. (The file contains the average price for each
year, the annual CPI, fuel consumption and distance
(thousands).) For each year calculate the inflationadjusted cost per year of driving. Use a graphical
technique to present the results.
3.40 Repeat Exercise 3.39 using the CPI-Annual file to
convert all amounts to constant 1982–1984 dollars.
Draw a line chart to show these data.
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
MPG
13.3
13.6
14.1
14.2
14.5
14.6
14.7
15.1
15.6
15.9
9.5
9.5
9.6
9.8
10.0
10.0
10.1
10.5
10.7
10.9
Year
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
MPG
16.4
16.9
16.9
16.7
16.7
16.8
16.9
17.0
16.9
17.7
Distance
11.1
11.3
11.6
11.6
11.7
11.8
11.8
12.1
12.2
12.2
Year
2000
2001
2002
2003
2004
2005
2006
MPG
16.9
17.1
16.9
17.0
17.1
17.2
17.1
Distance
12.2
11.9
12.2
12.2
12.2
12.1
12.4
Distance
Source: Statistical Abstract of the United States, 2009, Tables 1061 and 1062.
The following exercises require a computer and software.
3.42 Xr03-42 The monthly value of U.S. exports to
Canada (in $millions) and imports from Canada
from 1985 to 2009 were recorded. (Source: Federal
Reserve Economic Data.)
a.
b.
c.
d.
Draw a line chart of U.S. exports to Canada.
Draw a line chart of U.S. imports from Canada.
Calculate the trade balance and draw a line chart.
What do all the charts reveal?
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3.43 Xr03-43 The monthly value of U.S. exports to Japan
3.45 Xr03-45 The value of the Japanese yen in U.S.
(in $ millions) and imports from Japan from 1985 to
2009 were recorded. (Source: Federal Reserve
Economic Data.)
dollars was recorded monthly for the period
1971 to 2009. Draw a graph of these figures and
interpret your findings.
a.
b.
c.
d.
Draw a line chart of U.S. exports to Japan.
Draw a line chart of U.S. imports from Japan.
Calculate the trade balance and draw a line chart.
What do all the charts reveal?
3.46 Xr03-46 The Dow Jones Industrial Average was
3.44 Xr03-44 The value of the Canadian dollar in U.S. dol-
3.47 Refer to Exercise 3.46. Use the CPI-monthly file
lars was recorded monthly for the period 1971 to
2009. Draw a graph of these figures and interpret
your findings.
to measure the Dow Jones Industrial Average in
1982–1984 constant dollars. What have you
learned?
3 . 3 DESCRIBING
VA R I A B L E S
THE
recorded monthly for the years 1950 to 2009.
Use a graph to describe these numbers. (Source:
Wall Street Journal.)
R E L AT I O N S H I P
BET WEEN
T WO I N T E RVA L
Statistics practitioners frequently need to know how two interval variables are related.
For example, financial analysts need to understand how the returns of individual stocks
are related to the returns of the entire market. Marketing managers need to understand
the relationship between sales and advertising. Economists develop statistical techniques to describe the relationship between such variables as unemployment rates and
inflation. The technique is called a scatter diagram.
To draw a scatter diagram, we need data for two variables. In applications where
one variable depends to some degree on the other variable, we label the dependent variable Y and the other, called the independent variable, X. For example, an individual’s
income depends somewhat on the number of years of education. Accordingly, we identify income as the dependent variable and label it Y, and we identify years of education
as the independent variable and label it X. In other cases where no dependency is evident, we label the variables arbitrarily.
E X A M PL E 3.7
DATA
Xm03-07
Analyzing the Relationship between Price and
Size of Houses
A real estate agent wanted to know to what extent the selling price of a home is related to
its size. To acquire this information, he took a sample of 12 homes that had recently sold,
recording the price in thousands of dollars and the size in square feet. These data are
listed in the accompanying table. Use a graphical technique to describe the relationship
between size and price.
Size (ft2)
2,354
1,807
2,637
2,024
2,241
1,489
3,377
2,825
2,302
2,068
2,715
1,833
Price ($1,000)
315
229
355
261
234
216
308
306
289
204
265
195
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75
SOLUTION
Using the guideline just stated, we label the price of the house Y (dependent variable)
and the size X (independent variable). Figure 3.11 depicts the scatter diagram.
3.11
FIGURE
Scatter Diagram for Example 3.7
Scatter Diagram
400
350
Price
300
250
200
150
100
50
0
0
500
1000
1500
2000
Size
2500
3000
3500
4000
EXCEL
Scatter Diagram
400
350
Price
300
250
200
150
100
50
0
0
500
1000
1500
2000
Size
2500
3000
3500
4000
INSTRUCTIONS
1. Type or import the data into two adjacent columns. Store variable X in the first column and variable Y in the next column. (Open Xm03-07.)
2. Click Insert and Scatter.
3. To make cosmetic changes, click Chart Tools and Layout. (We chose to add titles
and remove the gridlines.) If you wish to change the scale, click Axes, Primary
Horizontal Axis or Primary Vertical Axis, More Primary Horizontal or Vertical
Axis Options . . . , and make the changes you want.
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MINITAB
Scatterplot of Price vs Size
360
340
320
Price
300
280
260
240
220
200
1500
2000
2500
Size
3000
3500
INSTRUCTIONS
1. Type or import the data into two columns. (Open Xm03-07.)
2. Click Graph and Scatterplot. . . .
3. Click Simple.
4. Type or use the Select button to specify the variable to appear on the Y-axis (Price)
and the X-axis (Size).
INTERPRET
The scatter diagram reveals that, in general, the greater the size of the house, the
greater the price. However, there are other variables that determine price. Further
analysis may reveal what these other variables are.
Patterns of Scatter Diagrams
As was the case with histograms, we frequently need to describe verbally how two variables are related. The two most important characteristics are the strength and direction
of the linear relationship.
Linearity
To determine the strength of the linear relationship, we draw a straight line through the
points in such a way that the line represents the relationship. If most of the points fall
close to the line, we say that there is a linear relationship. If most of the points appear
to be scattered randomly with only a semblance of a straight line, there is no, or at best,
a weak linear relationship. Figure 3.12 depicts several scatter diagrams that exhibit
various levels of linearity.
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77
In drawing the line freehand, we would attempt to draw it so that it passes through the middle of the data. Unfortunately, different people drawing a straight line through the same set
of data will produce somewhat different lines. Fortunately, statisticians have produced an
objective way to draw the straight line. The method is called the least squares method, and it
will be presented in Chapter 4 and employed in Chapters 16, 17, and 18.
FIGURE
3.12
Scatter Diagrams Depicting Linearity
y
x
(a) Strong linear relationship
y
x
(b) Medium-strength linear relationship
y
x
(c) Weak linear relationship
Note that there may well be some other type of relationship, such as a quadratic or
exponential one.
Direction
In general, if one variable increases when the other does, we say that there is a positive
linear relationship. When the two variables tend to move in opposite directions, we
describe the nature of their association as a negative linear relationship. (The terms
positive and negative will be explained in Chapter 4.) See Figure 3.13 for examples of
scatter diagrams depicting a positive linear relationship, a negative linear relationship,
no relationship, and a nonlinear relationship.
Interpreting a Strong Linear Relationship
In interpreting the results of a scatter diagram it is important to understand that if two variables are linearly related it does not mean that one is causing the other. In fact, we can never
conclude that one variable causes another variable. We can express this more eloquently as
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Correlation is not causation.
Now that we know what to look for, we can answer the chapter-opening example.
FIGURE
3.13
Scatter Diagrams Describing Direction
y
x
(a) Positive linear relationship
y
(b) Negative linear relationship
x
y
(c) No relationship
x
y
x
Were Oil Companies Gouging Customers
2000–2009: Solution
To determine whether drivers’ perceptions that oil companies were gouging consumers, we need
to determine whether and to what extent the two variables are related. The appropriate statistical
technique is the scatter diagram.
We label the price of gasoline Y and the price of oil X. Figure 3.14 displays the scatter
© Comstock Images/Jupiterimages
(d) Nonlinear relationship
diagram.
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FIGURE
3.14
79
Scatter Diagram for Chapter-Opening Example
Scatter Diagram
450
Price of gasoline
400
350
300
250
200
150
100
50
0
0
20
40
60
80
100
Price of Oil
120
140
160
EXCEL
Scatter Diagram
450
Price of gasoline
400
350
300
250
200
150
100
50
0
0
20
40
60
80
100
Price of Oil
120
140
160
MINITAB
Scatterplot of Price of gasoline vs Price of Oil
450
Price of gasoline
400
350
300
250
200
150
100
20
40
60
80
100
120
140
Price of Oil
(Continued)
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INTERPRET
The scatter diagram reveals that the two prices are strongly related linearly. When the price of oil was below $40,
the relationship between the two was stronger than when the price of oil exceeded $40.
We close this section by reviewing the factors that identify the use of the scatter
diagram.
Factors That Identify When to Use a Scatter Diagram
1. Objective: Describe the relationship between two variables
2. Data type: Interval
EXERCISES
3.48 Xr03-48 Between 2002 and 2005, there was a decrease
in movie attendance. There are several reasons for
this decline. One reason may be the increase in
DVD sales. The percentage of U.S. homes with
DVD players and the movie attendance (billions) in
the United States for the years 2000 to 2005 are
shown next. Use a graphical technique to describe
the relationship between these variables.
Year
2000 2001 2002
DVD percentage
12
Movie attendance 1.41
23
1.49
37
1.63
65
74
58
72
93
84
68
71
74
68
81
85
58 85
63 73
42
1.58
Calculus
Statistics
88
79
75
65
63
62
79
71
80
74
54
68
72
73
59
1.53
74
1.40
power of the dollar, investors seek investments that
will provide higher returns when inflation is higher.
It is frequently stated that common stocks provide
just such a hedge against inflation. The annual percentage rates of return on common stock and annual
inflation rates for a recent 10-year period are listed
here.
2
3
4
5
6
Returns
Inflation
25 8
6 11 21 -15
4.4 4.2 4.1 4.0 5.2 5.0
site for the statistics course, a sample of 15 students
was drawn. The marks for calculus and statistics
were recorded for each student. The data are as follows:
Calculus
Statistics
3.49 Xr03-49 Because inflation reduces the purchasing
1
3.50 Xr03-50 In a university where calculus is a prerequi-
2003 2004 2005
Sources: Northern Technology & Telecom Research and Motion Picture Association.
Year
b. Does it appear that the returns on common
stocks and inflation are linearly related?
7
8
9
10
12 -1 33 0
3.8 2.1 1.7 0.2
a. Use a graphical technique to depict the relationship between the two variables.
a. Draw a scatter diagram of the data.
b. What does the graph tell you about the relationship between the marks in calculus and statistics?
3.51 Xr03-51 The cost of repairing cars involved in acci-
dents is one reason that insurance premiums are so
high. In an experiment, 10 cars were driven into a
wall. The speeds were varied between 2 and 20 mph.
The costs of repair were estimated and are listed
here. Draw an appropriate graph to analyze the relationship between the two variables. What does the
graph tell you?
Speed
Cost of Repair ($)
2
4
6
8 10 12
88 124 358 519 699 816
Speed
Cost of Repair ($)
14
16
18
905 1,521 1,888
20
2,201
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3.52 Xr03-52 The growing interest in and use of the
Internet have forced many companies into considering ways to sell their products on the Web.
Therefore, it is of interest to these companies to
determine who is using the Web. A statistics practitioner undertook a study to determine how education and Internet use are connected. She took a
random sample of 15 adults (20 years of age and
older) and asked each to report the years of education they had completed and the number of hours of
Internet use in the previous week. These data follow.
a. Employ a suitable graph to depict the data.
b. Does it appear that there is a linear relationship
between the two variables? If so, describe it.
Education
Internet use
11
10
11
5
8
0
13
14
17
24
11
0
11
15
Education
Internet use
19
20
13
10
15
5
9
8
15
12
15
15
11
0
11
12
3.53 Xr03-53 A statistics professor formed the opinion that
students who handed in quiz and exams early outperformed students who handed in their papers
later. To develop data to decide whether her opinion
is valid, she recorded the amount of time (in minutes) taken by students to submit their midterm tests
(time limit 90 minutes) and the subsequent mark for
a sample of 12 students.
Time
90 73 86 85 80 87 90 78 84 71 72 88
Mark
68 65 58 94 76 91 62 81 75 83 85 74
The following exercises require a computer and software.
3.54 Xr03-54 In an attempt to determine the factors that
affect the amount of energy used, 200 households
were analyzed. The number of occupants and the
amount of electricity used were measured for each
household.
a. Draw a graph of the data.
b. What have you learned from the graph?
3.55 Xr03-55 Many downhill skiers eagerly look forward
to the winter months and fresh snowfalls. However,
winter also entails cold days. How does the temperature affect skiers’ desire? To answer this question, a
local ski resort recorded the temperature for 50 randomly selected days and the number of lift tickets
they sold. Use a graphical technique to describe the
data and interpret your results.
3.56 Xr03-56 One general belief held by observers of the
business world is that taller men earn more money
than shorter men. In a University of Pittsburgh
study, 250 MBA graduates, all about 30 years old,
were polled and asked to report their height (in
inches) and their annual income (to the nearest
$1,000).
81
a. Draw a scatter diagram of the data.
b. What have you learned from the scatter diagram?
3.57 Xr03-57 Do chief executive officers (CEOs) of pub-
licly traded companies earn their compensation?
Every year the National Post’s Business magazine
attempts to answer the question by reporting the
CEO’s annual compensation ($1,000), the profit (or
loss) ($1,000), and the three-year share return (%)
for the top 50 Canadian companies. Use a graphical
technique to answer the question.
3.58 Xr03-58 Are younger workers less likely to stay with
their jobs? To help answer this question, a random
sample of workers was selected. All were asked to
report their ages and how many months they had
been employed with their current employers. Use a
graphical technique to summarize these data.
(Adapted from Statistical Abstract of the United States,
2006, Table 599.)
3.59 Xr03-59 A very large contribution to profits for a
movie theater is the sales of popcorn, soft drinks,
and candy. A movie theater manager speculated that
the longer the time between showings of a movie,
the greater the sales of concession items. To acquire
more information, the manager conducted an experiment. For a month he varied the amount of time
between movie showings and calculated the sales.
Use a graphical technique to help the manager
determine whether longer time gaps produces
higher concession stand sales.
3.60 Xr03-60 An analyst employed at a commodities
trading firm wanted to explore the relationship
between prices of grains and livestock.
Theoretically, the prices should move in the same
direction because, as the price of livestock
increases, more livestock are bred, resulting in a
greater demand for grains to feed them. The analyst recorded the monthly grains and livestock
subindexes for 1971 to 2008. (Subindexes are
based on the prices of several similar commodities.
For example, the livestock subindex represents the
prices of cattle and hogs.) Using a graphical technique, describe the relationship between the two
subindexes and report your findings. (Source:
Bridge Commodity Research Bureau.)
3.61 Xr03-61 It is generally believed that higher interest
rates result in less employment because companies
are more reluctant to borrow to expand their business. To determine whether there is a relationship
between bank prime rate and unemployment, an
economist collected the monthly prime bank rate
and the monthly unemployment rate for the years
1950 to 2009. Use a graphical technique to supply
your answer. (Source: Bridge Commodity Research
Bureau.)
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A M E R I C A N N AT I O N A L E L E C T I O N S U R V E Y E X E R C I S E S
3.62
ANES2004* Do younger people have more education
(EDUC) than older people (AGE)? Use the
American National Election Survey from 2004 and a
graphical technique to help answer the question.
3.64 ANES2008* Analyze the relationship between the
In the 2008 survey American adults were asked to
report the amount of time (in minutes) that each person spent in an average day watching, reading, or listening about news in four different media. They are
3.65 ANES2008* Refer to Exercise 3.64. Study the scatter
Internet (TIME1)
Television (TIME2)
Printed newspaper (TIME3)
Radio (TIME4)
amount time reading news on the Internet and reading news in a printed newspaper. Does it appear that
they are linearly related?
diagram. Does it appear that something is wrong
with the data? If so, how do you correct the problem and determine whether a linear relationship
exists?
3.66 ANES2008* Graphically describe the relationship
3.63 ANES2008* Use a graphical technique to determine
whether people who spend more time reading news
on the Internet also devote more time to watching
news on television.
between the amount of time watching news on television and listening to news on the radio. Are the
two linearly related?
3.67 ANES2008* Do younger people spend more time read-
ing news on the Internet than older people? Use a
graphical technique to help answer the question.
GENR AL SO CIAL SURVEY EXERCISES
3.68 GSS2008* Do more educated people tend to marry
3.70 GSS2008* Is there a positive linear relationship
people with more education? Draw a scatter diagram of EDUC and SPEDUC to answer the
question.
between the amount of education of mothers (MAEDUC) and their children (EDUC)? Draw a scatter
diagram to answer the question.
3.69 GSS2008* Do the children of more educated men
3.71 GSS2008* If one member of a married couple works
(PAEDUC) have more education (EDUC)?
Produce a graph that helps answer the question.
more hours (HRS) does his or her spouse work less
hours (SPHRS)? Draw a graph to produce the information you need.
3 . 4 A RT
AND
SCIENCE
OF
G R A PH I C A L P R E S E N TAT I O N S
In this chapter and in Chapter 2, we introduced a number of graphical techniques. The
emphasis was on how to construct each one manually and how to command the computer to draw them. In this section, we discuss how to use graphical techniques effectively. We introduce the concept of graphical excellence, which is a term we apply to
techniques that are informative and concise and that impart information clearly to their
viewers. Additionally, we discuss an equally important concept: graphical integrity and
its enemy graphical deception.
Graphical Excellence
Graphical excellence is achieved when the following characteristics apply.
1.
The graph presents large data sets concisely and coherently. Graphical tech-
niques were created to summarize and describe large data sets. Small data sets are easily summarized with a table. One or two numbers can best be presented in a sentence.
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2.
83
The ideas and concepts the statistics practitioner wants to deliver are
clearly understood by the viewer. The chart is designed to describe what
would otherwise be described in words. An excellent chart is one that can replace
a thousand words and still be clearly comprehended by its readers.
3.
The graph encourages the viewer to compare two or more variables.
4.
The display induces the viewer to address the substance of the data and not
the form of the graph. The form of the graph is supposed to help present the sub-
5.
Graphs displaying only one variable provide very little information. Graphs are
often best used to depict relationships between two or more variables or to
explain how and why the observed results occurred.
stance. If the form replaces the substance, the chart is not performing its function.
There is no distortion of what the data reveal. You cannot make statistical
techniques say whatever you like. A knowledgeable reader will easily see through
distortions and deception. We will endeavor to make you a knowledgeable reader
by describing graphical deception later in this section.
Edward Tufte, professor of statistics at Yale University, summarized graphical
excellence this way:
1. Graphical excellence is the well-designed presentation of interesting data—
a matter of substance, of statistics, and of design.
2. Graphical excellence is that which gives the viewer the greatest number of
ideas in the shortest time with the least ink in the smallest space.
3. Graphical excellence is nearly always multivariate.
4. And graphical excellence requires telling the truth about the data.
Now let’s examine the chart that has been acclaimed the best chart ever drawn.
Figure 3.15 depicts Minard’s graph. The striped band is a time series depicting the
size of the army at various places on the map, which is also part of the chart. When
FIGURE
3.15
Chart Depicting Napoleon’s Invasion and Retreat from Russia
in 1812
CARTE FIGURATIVE des pertes successives en hommes de l'Armée Française dans la campagne de Russie 1812-1813.
Dressée par M.Minard, Inspecteur Général des Ponts et Chaussées en retraite.
TABLEAU GRAPHIQUE de la température en degrés du thermomètre de Réaumur au dessous de zéro
Source : Edward Tufte, The Visual Display of Quantitative Information (Cheshire, CT: Graphics Press, 1983), p. 41.
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Napoleon invaded Russia by crossing the Niemen River on June 21, 1812, there were
422,000 soldiers. By the time the army reached Moscow, the number had dwindled to
100,000. At that point, the army started its retreat. The black band represents the army
in retreat. At the bottom of the chart, we see the dates starting with October 1813. Just
above the dates, Minard drew another time series, this one showing the temperature. It
was bitterly cold during the fall, and many soldiers died of exposure. As you can see, the
temperature dipped to –30 on December 6. The chart is effective because it depicts five
variables clearly and succinctly.
Graphical Deception
The use of graphs and charts is pervasive in newspapers, magazines, business and economic reports, and seminars, in large part because of the increasing availability of computers and software that allow the storage, retrieval, manipulation, and summary of
large masses of raw data. It is therefore more important than ever to be able to evaluate
critically the information presented by means of graphical techniques. In the final
analysis, graphical techniques merely create a visual impression, which is easy to distort.
In fact, distortion is so easy and commonplace that in 1992 the Canadian Institute of
Chartered Accountants found it necessary to begin setting guidelines for financial
graphics, after a study of hundreds of the annual reports of major corporations found
that 8% contained at least one misleading graph that covered up bad results. Although
the heading for this section mentions deception, it is quite possible for an inexperienced
person inadvertently to create distorted impressions with graphs. In any event, you
should be aware of possible methods of graphical deception. This section illustrates a
few of them.
The first thing to watch for is a graph without a scale on one axis. The line chart of
a firm’s sales in Figure 3.16 might represent a growth rate of 100% or 1% over the
5 years depicted, depending on the vertical scale. It is best simply to ignore such graphs.
3.16
Graph without Scale
Sales
FIGURE
1996
1997
1998
1999
2000
A second trap to avoid is being influenced by a graph’s caption. Your impression of
the trend in interest rates might be different, depending on whether you read a newspaper carrying caption (a) or caption (b) in Figure 3.17.
Perspective is often distorted if only absolute changes in value, rather than percentage changes, are reported. A $1 drop in the price of your $2 stock is relatively
more distressing than a $1 drop in the price of your $100 stock. On January 9, 1986,
newspapers throughout North America displayed graphs similar to the one shown in
Figure 3.18 and reported that the stock market, as measured by the Dow Jones
Industrial Average (DJIA), had suffered its worst 1-day loss ever on the previous day.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
FIGURE
3.17
Graphs with Different Captions
12
Interest rate
(percent)
12
Interest rate
(percent)
85
11
10
9
8
11
10
9
8
2
9
16
July
23
30
(a) Interest rates have finally begun
to turn downward.
2
9
16
July
23
30
(b) Last week provided temporary relief from the
upward trend in interest rates.
The loss was 39 points, exceeding even the loss of Black Tuesday: October 28, 1929.
While the loss was indeed a large one, many news reports failed to mention that the
1986 level of the DJIA was much higher than the 1929 level. A better perspective on
the situation could be gained by noticing that the loss on January 8, 1986, represented a 2.5% decline, whereas the decline in 1929 was 12.8%. As a point of interest,
we note that the stock market was 12% higher within 2 months of this historic drop
and 40% higher 1 year later. The largest one-day percentage drop in the DJIA is
24.4% (December 12, 1914).
FIGURE
3.18
Graph Showing Drop in the DJIA
1570
DJIA
1560
1550
1540
1530
1520
Jan. 3
Jan. 6
Jan. 7
Jan. 8
Jan. 9
We now turn to some rather subtle methods of creating distorted impressions with
graphs. Consider the graph in Figure 3.19, which depicts the growth in a firm’s quarterly sales during the past year, from $100 million to $110 million. This 10% growth in
quarterly sales can be made to appear more dramatic by stretching the vertical axis—a
technique that involves changing the scale on the vertical axis so that a given dollar
amount is represented by a greater height than before. As a result, the rise in sales
appears to be greater because the slope of the graph is visually (but not numerically)
steeper. The expanded scale is usually accommodated by employing a break in the vertical axis, as in Figure 3.20(a), or by truncating the vertical axis, as in Figure 3.20(b), so
that the vertical scale begins at a point greater than zero. The effect of making slopes
appear steeper can also be created by shrinking the horizontal axis, in which case points
on the horizontal axis are moved closer together.
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CHAPTER 3
3.19
FIGURE
Graph Showing Growth in Quarterly Sales 1
Sales
(millions of dollars)
125
100
75
50
25
0
Mar.
Sept.
June
Dec.
Quarter
3.20
Graph Showing Growth in Quarterly Sales 2
110
Sales
(millions of dollars)
Sales
(millions of dollars)
FIGURE
108
106
104
102
100
0
Mar.
June Sept.
Quarter
110
108
106
104
102
100
98
Mar.
Dec.
(a) Break in vertical axis
June Sept.
Quarter
Dec.
(b) Truncated vertical axis
Just the opposite effect is obtained by stretching the horizontal axis; that is, spreading out the points on the horizontal axis to increase the distance between them so that
slopes and trends will appear to be less steep. The graph of a firm’s profits presented in
Figure 3.21(a) shows considerable swings, both upward and downward in the profits
from one quarter to the next. However, the firm could convey the impression of reasonable stability in profits from quarter to quarter by stretching the horizontal axis, as
shown in Figure 3.21(b).
FIGURE
3.21
Graph Showing Considerable Swings or Relative Stability
50
Profits
(millions of dollars)
Profits
(millions of dollars)
50
40
30
20
10
12341234
Quarter
(a) Compressed horizontal axis
40
30
20
10
1
2
3
4
1
2
3
4
Quarter
(b) Stretched horizontal axis
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
87
Similar illusions can be created with bar charts by stretching or shrinking the vertical
or horizontal axis. Another popular method of creating distorted impressions with bar
charts is to construct the bars so that their widths are proportional to their heights. The
bar chart in Figure 3.22(a) correctly depicts the average weekly amount spent on food by
Canadian families during three particular years. This chart correctly uses bars of equal
width so that both the height and the area of each bar are proportional to the expenditures
they represent. The growth in food expenditures is exaggerated in Figure 3.22(b), in
which the widths of the bars increase with their heights. A quick glance at this bar chart
might leave the viewer with the mistaken impression that food expenditures increased
fourfold over the decade, because the 1995 bar is four times the size of the 1985 bar.
FIGURE
3.22
Correct and Distorted Bar Charts
120
Dollars
Dollars
120
60
0
1985
1990
1995
60
0
1985
1990
1995
Year
Year
(a) Correct bar chart
(b) Increasing bar widths to create
distortion
You should be on the lookout for size distortions, particularly in pictograms,
which replace the bars with pictures of objects (such as bags of money, people, or
animals) to enhance the visual appeal. Figure 3.23 displays the misuse of a pictogram—
the snowman grows in width as well as height. The proper use of a pictogram is shown
in Figure 3.24, which effectively uses pictures of Coca-Cola bottles.
FIGURE
3.23
Misuse of Pictogram
Snowfall in Metro climbs relentlessly
Snowfall last winter was more than
50% greater than the previous winter,
and more than double what fell four
winters ago.
1992–93
1988–89
79.8 cm
1991–92
95.2 cm
163.6 cm
The preceding examples of creating a distorted impression using graphs are not
exhaustive, but they include some of the more popular methods. They should also serve
to make the point that graphical techniques are used to create a visual impression, and
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CHAPTER 3
FIGURE
3.24
Correct Pictogram
Shareholders Get More
for Their Money
Return on Coca-Cola’s
shareholders’ equity,
in percent.
29.5%
22.1%
15.3%
9.7%
’85
’87
’91
’92
the impression you obtain may be a distorted one unless you examine the graph with
care. You are less likely to be misled if you focus your attention on the numerical values
that the graph represents. Begin by carefully noting the scales on both axes; graphs with
unmarked axes should be ignored completely.
EXERCISES
3.72 Xr03-72 A computer company has diversified its
operations into financial services, construction, manufacturing, and hotels. In a recent annual report, the
following tables were provided. Create charts to present these data so that the differences between last
year and the previous year are clear. (Note: It may be
necessary to draw the charts manually.)
Sales (Millions of Dollars)
by Region
Region
United States
Canada
Europe
Australasia
Total
Last Year
67.3
20.9
37.9
26.2
152.2
Previous Year
40.4
18.9
35.5
10.3
105.1
Sales (Millions of Dollars) by
Division
Division
Customer service
Library systems
Last Year
Previous Year
54.6
49.3
43.8
30.5
Construction/property
management
17.5
Manufacturing and
distribution
15.4
Financial systems
9.4
Hotels and clubs
5.9
7.7
8.9
10.9
3.4
3.73 Xr03-73 The following table lists the number (in
thousands) of violent crimes and property crimes
committed annually in 1985 to 2006 (the last year
data were available).
a. Draw a chart that displays both sets of data.
b. Does it appear that crime rates are decreasing?
Explain.
c. Is there another variable that should be included
to show the trends in crime rates?
Year
1985
1986
1987
1988
Violent crimes
1,328
1,489
1,484
1,566
Property crimes
11,103
11,723
12,025
12,357
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
1,646
1,820
1,912
1,932
1,926
1,858
1,799
1,689
1,636
1,534
1,426
1,425
1,439
1,424
1,384
1,360
1,391
1,418
12,605
12,655
12,961
12,506
12,219
12,132
12,064
11,805
11,558
10,952
10,208
10,183
10,437
10,455
10,443
10,319
10,175
9,984
Source: Statistical Abstract of the United States, 2006, Table 293; and 2009,
Table 293.
3.74 Xr03-74 Refer to Exercise 3.73. We’ve added the
United States population.
a. Incorporate this variable into your charts to show
crime rate trends.
b. Summarize your findings.
c. Can you think of another demographic variable
that may explain crime rate trends?
Refer to Exercises 3.73 and 3.74.
We’ve included the number of Americans aged
15 to 24.
3.75 Xr03-75
a. What is the significance of adding the populations aged 15 to 24?
b. Include these data in your analysis. What have
you discovered?
3.76 Xr03-76 To determine premiums for automobile
insurance, companies must have an understanding of
the variables that affect whether a driver will have an
accident. The age of the driver may top the list of
variables. The following table lists the number of
drivers in the United States, the number of fatal
accidents, and the number of total accidents in each
age group in 2002.
a. Calculate the accident rate (per driver) and the
fatal accident rate (per 1,000 drivers) for each age
group.
b. Graphically depict the relationship between the
ages of drivers, their accident rates, and their
fatal accident rates (per 1,000 drivers).
c. Briefly describe what you have learned.
Number of
Drivers
Age Group (1,000s)
Under 20
20–24
25–34
35–44
45–54
55–64
65–74
Over 74
Total
9,508
16,768
33,734
41,040
38,711
25,609
15,812
12,118
193,300
89
Number of
Accidents
(1,000s)
Number of
Fatal
Accidents
3,543
2,901
7,061
6,665
5,136
2,775
1,498
1,121
30,700
6,118
5,907
10,288
10,309
8,274
5,322
2,793
3,689
52,700
Source: National Safety Council.
3.77 Xr03-77 During 2002 in the state of Florida, a total of
365,474 drivers were involved in car accidents. The
accompanying table breaks down this number by the
age group of the driver and whether the driver was
injured or killed. (There were actually 371,877 accidents, but the driver’s age was not recorded in 6,413
of these.)
a. Calculate the injury rate (per 100 accidents) and
the death rate (per accident) for each age group.
b. Graphically depict the relationship between the
ages of drivers, their injury rate (per 100 accidents), and their death rate.
c. Briefly describe what you have learned from
these graphs.
d. What is the difference between the information
extracted from Exercise 3.9 and this one?
Number of
Age Group Accidents
Drivers
Injured
Drivers
Killed
20 or less
52,313
21–24
38,449
25–34
78,703
35–44
76,152
45–54
54,699
55–64
31,985
65–74
18,896
75–84
11,526
85 or more
2,751
Total
365,474
21,762
16,016
31,503
30,542
22,638
13,210
7,892
5,106
1,223
149,892
217
185
324
389
260
167
133
138
65
1,878
Source: Florida Department of Highway Safety and Motor Vehicles.
3.78 Xr03-78 The accompanying table lists the average test
scores in the Scholastic Assessment Test (SAT) for
the years 1967, 1970, 1975, 1980, 1985, 1990, 1995,
and 1997 to 2007.
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CHAPTER 3
Year
1967
1970
1975
1980
1985
1990
1995
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Verbal All
543
537
512
502
509
500
504
505
505
505
505
506
504
507
508
508
503
502
Verbal Male
Verbal Female
540
536
515
506
514
505
505
507
509
509
507
509
507
512
512
513
505
504
545
538
509
498
503
496
502
503
502
502
504
502
502
503
504
505
502
502
Math All
516
512
498
492
500
501
506
511
512
511
514
514
516
519
518
520
518
515
Math Male
Math Female
535
531
518
515
522
521
525
530
531
531
533
533
534
537
537
538
536
533
595
493
479
473
480
483
490
494
496
495
498
498
500
503
501
504
502
499
Source: Statistical Abstract of the United States, 2003, Table 264; 2006, Table 252; 2009, Table 258.
Draw a chart for each of the following.
a. You wish to show that both verbal and mathematics test scores for all students have not
changed much over the years.
b. The exact opposite of part (a).
c. You want to claim that there are no differences
between genders.
d. You want to “prove” that differences between
genders exist.
b. Draw a bar chart of these data with 0.0% as the
lowest point on the vertical axis.
c. Discuss the impression given by the two charts.
d. Which chart would you use? Explain.
Month 1 2
3 4 5 6 7 8 9 10 11 12
Rate 7.5 7.6 7.5 7.3 7.2 7.1 7.0 6.7 6.4 6.5 6.3 6.0
3.80 Xr03-80 The accompanying table lists the federal mini-
mum wage from 1955 to 2007. The actual and adjusted
minimum wages (in constant 1996 dollars) are listed.
3.79 Xr03-79 The monthly unemployment rate in one
a. Suppose you wish to show that the federal minimum wage has grown rapidly over the years.
Draw an appropriate chart.
b. Draw a chart to display the actual changes in the
federal minimum wage.
state for the past 12 months is listed here.
a. Draw a bar chart of these data with 6.0% as the
lowest point on the vertical axis.
Year
Current
Dollars
Constant
1996
Dollars
Year
Current
Dollars
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
0.75
1.00
1.00
1.00
1.00
1.00
1.15
1.15
1.25
1.25
1.25
1.25
1.40
4.39
5.77
5.58
5.43
5.39
5.30
6.03
5.97
6.41
6.33
6.23
6.05
6.58
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
3.35
3.35
3.35
3.35
3.35
3.35
3.35
3.35
3.80
4.25
4.25
4.25
4.25
Constant
1996
Dollars
5.78
5.28
5.06
4.88
4.80
4.63
4.44
4.24
4.56
4.90
4.75
4.61
4.50
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1.60
1.60
1.60
1.60
1.60
1.60
2.00
2.10
2.30
2.30
2.65
2.90
3.10
3.35
7.21
6.84
6.47
6.20
6.01
5.65
6.37
6.12
6.34
5.95
6.38
6.27
5.90
5.78
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
4.25
4.75
5.15
5.15
5.15
5.15
5.15
5.15
5.15
5.15
5.15
5.15
5.85
91
4.38
4.75
5.03
4.96
4.85
4.69
4.56
4.49
4.39
4.28
4.14
4.04
4.41
Source: U.S. Department of Labor.
3.81 Xr03-81 The following table shows school enrollment
(in thousands) for public and private schools for the
years 1965 to 2005.
a. Draw charts that allow you to claim that enrollment in private schools is “skyrocketing.”
b. Draw charts that “prove” public school enrollment is stagnant.
Year
Public_K_8
Private_K_8
Public_9_12
Private_9_12
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
30,563
31,145
31,641
32,226
32,513
32,558
32,318
31,879
31,401
30,971
30,515
29,997
29,375
28,463
28,034
27,647
27,280
27,161
26,981
26,905
27,034
27,420
27,933
28,501
29,152
29,878
30,506
31,088
31,504
31,898
32,341
32,764
4,900
4,800
4,600
4,400
4,200
4,052
3,900
3,700
3,700
3,700
3,700
3,825
3,797
3,732
3,700
3,992
4,100
4,200
4,315
4,300
4,195
4,116
4,232
4,036
4,035
4,084
4,518
4,528
4,536
4,624
4,721
4,720
11,610
11,894
12,250
12,718
13,037
13,336
13,753
13,848
14,044
14,103
14,304
14,314
14,203
14,088
13,616
13,231
12,764
12,405
12,271
12,304
12,388
12,333
12,076
11,687
11,390
11,338
11,541
11,735
11,961
12,213
12,500
12,847
1,400
1,400
1,400
1,400
1,300
1,311
1,300
1,300
1,300
1,300
1,300
1,342
1,343
1,353
1,300
1,339
1,400
1,400
1,400
1,400
1,362
1,336
1,247
1,206
1,163
1,150
1,163
1,148
1,132
1,162
1,197
1,213
College_Public College_Private
3,970
4,349
4,816
5,431
5,897
6,428
6,804
7,071
7,420
7,989
8,835
8,653
8,847
8,786
9,037
9,457
9,647
9,696
9,683
9,477
9,479
9,714
9,973
10,161
10,578
10,845
11,310
11,385
11,189
11,134
11,092
11,121
1,951
2,041
2,096
2,082
2,108
2,153
2,144
2,144
2,183
2,235
2,350
2,359
2,439
2,474
2,533
2,640
2,725
2,730
2,782
2,765
2,768
2,790
2,793
2,894
2,961
2,974
3,049
3,102
3,116
3,145
3,169
3,247
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CHAPTER 3
1997
1998
1999
2000
2001
2002
2003
2004
2005
33,073
33,346
33,488
33,688
33,938
34,116
34,202
34,178
34,205
4,726
4,748
4,765
4,878
4,993
4,886
4,761
4,731
4,699
13,054
13,193
13,369
13,515
13,734
14,067
14,338
14,617
14,909
1,218
1,240
1,254
1,292
1,326
1,334
1,338
1,356
1,374
11,196
11,138
11,309
11,753
12,233
12,752
12,857
12,980
13,022
3,306
3,369
3,482
3,560
3,695
3,860
4,043
4,292
4,466
Source: Statistical Abstract of the United States, 2009, Table 211.
3.82 Xr03-82 The following table lists the percentage of
the home has not changed much in the past
47 years.
b. Use a chart to show that the percentage of
single women in the workforce has increased
“dramatically.”
single and married women in the United States who
had jobs outside the home during the period 1970 to
2007.
a. Construct a chart that shows that the percentage of married women who are working outside
Year
Single
Married
Year
Single
Married
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
56.8
56.4
57.5
58.6
59.5
59.8
61.0
62.1
63.7
64.6
64.4
64.5
65.1
65.0
65.6
66.6
67.2
67.4
67.7
40.5
40.6
41.2
42.3
43.3
44.3
45.3
46.4
47.8
49.0
49.8
50.5
51.1
51.8
52.8
53.8
54.9
55.9
56.7
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
68.0
66.7
66.2
66.2
66.2
66.7
66.8
67.1
67.9
68.5
68.7
68.9
68.1
67.4
66.2
65.9
66.0
65.7
65.3
57.8
58.4
58.5
59.3
59.4
60.7
61.0
61.2
61.6
61.2
61.2
61.1
61.2
61.0
61.0
60.5
60.7
61.0
61.0
Source: Statistical Abstract of the United States, 2009, Table 286.
CHAPTER SUM M ARY
Histograms are used to describe a single set of interval data.
Statistics practitioners examine several aspects of the shapes
of histograms. These are symmetry, number of modes, and
its resemblance to a bell shape.
We described the difference between time-series data
and cross-sectional data. Time series are graphed by line
charts.
To analyze the relationship between two interval variables, we draw a scatter diagram. We look for the direction
and strength of the linear relationship.
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
93
IMPORTANT TERMS
Classes 46
Histogram 46
Symmetry 50
Positively skewed 50
Negatively skewed 50
Modal class 50
Unimodal 50
Bimodal 51
Stem-and-leaf display 57
Depths 58
Relative frequency distribution 59
Cumulative relative frequency distribution 59
Ogive 60
Credit scorecard 63
Cross-sectional data 64
Time-series data 64
Line chart 65
Scatter diagram 74
Linear relationship 76
Positive linear relationship 77
Negative linear relationship 77
Graphical excellence 82
Graphical deception 82
COMPUTER OUTPUT AND INSTRUCTIONS
Graphical Technique
Histogram
Stem-and-leaf display
Ogive
Line chart
Scatter diagram
Excel
Minitab
47
58
60
66
75
48
58
61
67
76
CHAPTER EXERCISES
The following exercises require a computer and software.
3.83
Gold and other precious metals have traditionally been considered a hedge against inflation. If this is
true, we would expect that a fund made up of precious
metals (gold, silver, platinum, and others) would have a
strong positive relationship with the inflation rate. To
see whether this is true, a statistics practitioner collected the monthly CPI and the monthly precious metals subindex, which is based on the prices of gold,
silver, platinum, etc., for the years 1975 to 2008. These
figures were used to calculate the monthly inflation
rate and the monthly return on the precious metals
subindex. Use a graphical technique to determine the
nature of the relationship between the inflation rate
and the return on the subindex. What does the graph
tell you? (Source: U.S. Treasury and Bridge
Commodity Research Bureau.)
Xr03-83
3.84 Xr03-84 The monthly values of one Australian dollar
measured in American dollars since 1971 were
recorded. Draw a graph that shows how the
exchange rate has varied over the 38-year period.
(Source: Federal Reserve Economic Data.)
3.85 Xr03-85 Studies of twins may reveal more about the
“nature or nurture” debate. The issue being debated is
whether nature or the environment has more of an
effect on individual traits such as intelligence. Suppose
that a sample of identical twins was selected and their
IQs measured. Use a suitable graphical technique to
depict the data, and describe what it tells you about the
relationship between the IQs of identical twins.
3.86 Xr03-86 An economist wanted to determine whether
a relationship existed between interest rates and currencies (measured in U.S. dollars). He recorded the
monthly interest rate and the currency indexes for
the years 1982 to 2008. Graph the data and describe
the results. (Source: Bridge Commodity Research
Bureau.)
3.87 Xr03-87 One hundred students who had reported that
they use their computers for at least 20 hours per
week were asked to keep track of the number of
crashes their computers incurred during a 12-week
period. Using an appropriate statistical method,
summarize the data. Describe your findings.
3.88 Xr03-88 In Chapters 16, 17, and 18, we introduce
regression analysis, which addresses the relationships among variables. One of the first applications
of regression analysis was to analyze the relationship
between the heights of fathers and sons. Suppose
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that a sample of 80 fathers and sons was drawn. The
heights of the fathers and of the adult sons were
measured.
a. Draw a scatter diagram of the data. Draw a
straight line that describes the relationship.
b. What is the direction of the line?
c. Does it appear that there is a linear relationship
between the two variables?
3.89 Xr03-89 When the Dow Jones Industrial Averages
index increases, it usually means that the economy is
growing, which in turn usually means that the
unemployment rate is low. A statistics professor
pointed out that in numerous periods (including
when this edition was being written), the stock market had been booming while the rest of the economy
was performing poorly. To learn more about the
issue, the monthly closing DJIA and the monthly
unemployment rates were recorded for the years
1950 to 2009. Draw a graph of the data and report
your results. (Source: Federal Reserve Economic
Data and the Wall Street Journal.)
3.90 Xr03-90 The monthly values of one British pound
measured in American dollars since 1987 were
recorded. Produce a graph that shows how the
exchange rate has varied over the past 23 years.
(Source: Federal Reserve Economic Data.)
3.91 Xr03-91 Do better golfers play faster than poorer
ones? To determine whether a relationship exists, a
sample of 125 foursomes was selected. Their total
scores and the amount of time taken to complete the
18 holes were recorded. Graphically depict the data,
and describe what they tell you about the relationship between score and time.
3.92 Xr03-92 The value of monthly U.S. exports to
Mexico and imports from Mexico (in $ millions)
since 1985 were recorded. (Source: Federal Reserve
Economic Data.)
a. Draw a chart that depicts exports.
b. Draw a chart that exhibits imports.
c. Compute the trade balance and graph these data.
d. What do these charts tell you?
3.93 Xr03-93 An increasing number of consumers prefer
to use debit cards in place of cash or credit cards. To
analyze the relationship between the amounts of
purchases made with debit and credit cards,
240 people were interviewed and asked to report the
amount of money spent on purchases using debit
cards and the amount spent using credit cards during the last month. Draw a graph of the data and
summarize your findings.
3.94 Xr03-94 Most publicly traded companies have boards
of directors. The rate of pay varies considerably.
A survey was undertaken by the Globe and Mail
(February 19, 2001) wherein 100 companies were
surveyed and asked to report how much their directors were paid annually. Use a graphical technique to
present these data.
3.95 Xr03-95 Refer to Exercise 3.94. In addition to reporting
the annual payment per director, the survey recorded
the number of meetings last year. Use a graphical technique to summarize and present these data.
3.96 Xr03-96 Is airline travel becoming safer? To help
answer this question, a student recorded the number
of fatal accidents and the number of deaths that
occurred in the years 1986 to 2007 for scheduled airlines. Use a graphical method to answer the question. (Source: Statistical Abstract of the United States,
2009, Table 1036.)
3.97 Xr03-97 Most car-rental companies keep their cars
for about a year and then sell them to used car dealerships. Suppose one company decided to sell the
used cars themselves. Because most used car buyers
make their decision on what to buy and how much
to spend based on the car’s odometer reading, this
would be an important issue for the car-rental company. To develop information about the mileage
shown on the company’s rental cars, the general
manager took a random sample of 658 customers
and recorded the average number of miles driven
per day. Use a graphical technique to display these
data.
3.98 Xr03-98 Several years ago, the Barnes Exhibit toured
major cities all over the world, with millions of people flocking to see it. Dr. Albert Barnes was a
wealthy art collector who accumulated a large number of impressionist masterpieces; the total exceeds
800 paintings. When Dr. Barnes died in 1951, he
stated in his will that his collection was not to be
allowed to tour. However, because of the deterioration of the exhibit’s home near Philadelphia, a judge
ruled that the collection could go on tour to raise
enough money to renovate the building. Because of
the size and value of the collection, it was predicted
(correctly) that in each city a large number of people
would come to view the paintings. Because space
was limited, most galleries had to sell tickets that
were valid at one time (much like a play). In this way,
they were able to control the number of visitors at
any one time. To judge how many people to let in at
any time, it was necessary to know the length of time
people would spend at the exhibit; longer times
would dictate smaller audiences; shorter times
would allow for the sale of more tickets. The manager of a gallery that will host the exhibit realized
her facility can comfortably and safely hold about
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GRAPHICAL DESCRIPTIVE TECHNIQUES II
250 people at any one time. Although the demand
will vary throughout the day and from weekday to
weekend, she believes that the demand will not drop
below 500 at any time. To help make a decision
about how many tickets to sell, she acquired the
amount of time a sample of 400 people spent at the
exhibit from another city. What ticket procedure
should the museum management institute?
The following exercises are based on data sets that include additional
data referenced in previously presented examples and exercises.
95
3.100 Xm03-03* Xm03-04* In addition to the previously dis-
cussed data in Examples 3.3 and 3.4, the professor
listed the midterm mark. Conduct an analysis of the
relationship between final exam mark and midterm
mark in each course. What does this analysis tell
you?
3.101 Xr02-54* Two other questions were asked in
Exercise 2.54:
Number of weeks job searching?
Salary ($ thousands)?
3.99 Xm03-03* Xm03-04* Examples 3.3 and 3.4 listed the
final marks in the business statistics course and the
mathematical statistics course. The professor also
provided the final marks in the first-year required
calculus course. Graphically describe the relationship between calculus and statistics marks. What
information were you able to extract?
CASE 3.1
DATA
C03-01a
C03-01b
The placement office wants the following:
a. Graphically describe salary.
b. Is salary related to the number of weeks needed
to land the job?
The Question of Global Warming
I
n the last part of the 20th century,
If the planet is warming, is there a
vital to our life on Earth because it is
scientists developed the theory that
human cause or is it natural fluctu-
necessary for growing plants. The
the planet was warming and that
ation? Earth’s temperature has
amount of CO2 produced by fossil
the primary cause was the increasing
increased and decreased many
fuels is a relatively small proportion
amounts of atmospheric carbon dioxide
times in its long history. We’ve had
of all the CO2 in the atmosphere.
2.
(CO2), which are the product of burning
higher temperatures, and we’ve had
The generally accepted procedure is to
oil, natural gas, and coal (fossil fuels).
lower temperatures, including vari-
record monthly temperature anomalies.
Although many climatologists believe in
ous ice ages. In fact, a period called
To do so, we calculate the average for
the so-called greenhouse effect, many
the “Little Ice Age” ended around
each month over many years. We then
others do not subscribe to this theory.
the middle to the end of the 19th
calculate any deviations between the
There are three critical questions that
century. Then the temperature rose
latest month’s temperature reading and
need to be answered in order to resolve
until about 1940, at which point it
its average. A positive anomaly would
the issue.
decreased until 1975. In fact, an
represent a month’s temperature that is
April 28, 1975, Newsweek article
above the average. A negative anomaly
discussed the possibility of global
indicates a month where the tempera-
cooling, which seemed to be the
ture is less than the average. One key
consensus among scientists.
question is how we measure the
If the planet is warming, is CO2 the
temperature.
1.
Is Earth actually warming? To
answer this question, we need
accurate temperature measurements over a large number of years.
But how do we measure the temperature before the invention of
accurate thermometers? Moreover,
how do we go about measuring
Earth’s temperature even with
accurate thermometers?
3.
cause? There are greenhouse gases
in the atmosphere, without which
Earth would be considerably colder.
These gases include methane, water
vapor, and carbon dioxide. All occur
naturally in nature. Carbon dioxide is
Although there are many different
sources of data, we have chosen to provide you with one, the National Climatic
Data Center (NCDC), which is affiliated
with the National Oceanic and
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(Case 3.1 continued)
Atmospheric Administration (NOAA).
(Other sources tend to agree with the
NCDC’s data.) C03-01a stores the
monthly temperature anomalies from
1880 to 2009.
However, attempts to estimate CO2 lev-
(Note that some of the original data are
els prior to 1958 are as controversial as
missing and were replaced by interpo-
the methods used to estimate tempera-
lated values.)
tures. These techniques include taking
ice-core samples from the arctic and
trapped in the ice from which estimates
of atmospheric CO2 are produced. To
atmosphere come from the Mauna Loa
avoid this controversy, we will use the
Observatory in Hawaii, which has mea-
Mauna Loa Observatory numbers only.
sured this variable since December 1958.
These data are stored in file C03-01b.
Use whichever techniques you wish
to determine whether there is
measuring the amount of carbon dioxide
The best measures of CO2 levels in the
CASE 3.2
1.
global warming.
2.
Use a graphical technique to determine whether there is a relationship
between temperature anomalies
and CO2 levels.
Economic Freedom and Prosperity
A
subjective score for 10 freedoms: busidom, government size, monetary free-
sible to compare the GDP for all coun-
protect the liberty of individuals, greater
dom, investment freedom, financial
tries. The GDP PPP figures for 2008
prosperity results for all. Since 1995, the
freedom, property rights, freedom from
(the latest year available) are stored in
Wall Street Journal and the Heritage
corruption, and labor freedom. We
C03-02b. Use the 2009 Freedom Index
Foundation, a think tank in Washington,
downloaded the scores for the years
scores, the GDP PPP figures, and a
D.C., have produced the Index of
1995 to 2009 and stored them in
graphical technique to see how freedom
Economic Freedom for all countries in
C03-02a. From the CIA Factbook, we
and prosperity are related.
the world. The index is based on a
determined the gross domestic product
dam Smith published The Wealth
(GDP), measured in terms purchasing
of Nations in 1776. In that book
ness freedom, trade freedom, fiscal free-
power parity (PPP), which makes it pos-
he argued that when institutions
DATA
C03-02a
C03-02b
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4
© R.L./Shutterstock
NUMERICAL DESCRIPTIVE
TECHNIQUES
4.1
Measures of Central Location
4.2
Measures of Variability
4.3
Measures of Relative Standing and Box Plots
4.4
Measures of Linear Relationship
4.5
(Optional) Applications in Professional Sports: Baseball
4.6
(Optional) Applications in Finance: Market Model
4.7
Comparing Graphical and Numerical Techniques
4.8
General Guidelines for Exploring Data
Appendix 4
Review of Descriptive Techniques
DATA
Xm04-00
In the era of free agency, professional sports teams must compete for the services of
the best players. It is generally believed that only teams whose salaries place them in
the top quarter have a chance of winning the championship. Efforts have been made
to provide balance by establishing salary caps or some form of equalization. To examine the problem, we gathered data from the 2009 baseball season. For each team in major league baseball, we
© AP Photo/Charles Krupa
The Cost of One More Win in Major League Baseball
recorded the number of wins and the team payroll.
To make informed decisions, we need to know how the number of wins and the team payroll
are related. After the statistical technique is presented, we return to this problem and solve it.
97
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CHAPTER 4
I
I N T RO D U C T I O N
n Chapters 2 and 3, we presented several graphical techniques that describe data. In
this chapter we introduce numerical descriptive techniques that allow the statistics
practitioner to be more precise in describing various characteristics of a sample or
population. These techniques are critical to the development of statistical inference.
As we pointed out in Chapter 2, arithmetic calculations can be applied to interval
data only. Consequently, most of the techniques introduced here may be used only to
numerically describe interval data. However, some of the techniques can be used for
ordinal data, and one of the techniques can be employed for nominal data.
When we introduced the histogram, we commented that there are several bits of
information that we look for. The first is the location of the center of the data. In
Section 4.1, we will present measures of central location. Another important characteristic that we seek from a histogram is the spread of the data. The spread will be measured more precisely by measures of variability, which we present in Section 4.2.
Section 4.3 introduces measures of relative standing and another graphical technique,
the box plot.
In Section 3.3, we introduced the scatter diagram, which is a graphical method that
we use to analyze the relationship between two interval variables. The numerical counterparts to the scatter diagram are called measures of linear relationship, and they are presented in Section 4.4.
Sections 4.5 and 4.6 feature statistical applications in baseball and finance, respectively. In Section 4.7, we compare the information provided by graphical and numerical
techniques. Finally, we complete this chapter by providing guidelines on how to explore
data and retrieve information.
S A M PL E S TAT I S T I C
OR
P O P U L AT I O N PA R A M E T E R
Recall the terms introduced in Chapter 1: population, sample, parameter, and statistic.
A parameter is a descriptive measurement about a population, and a statistic is a
descriptive measurement about a sample. In this chapter, we introduce a dozen descriptive measurements. For each one, we describe how to calculate both the population
parameter and the sample statistic. However, in most realistic applications, populations
are very large—in fact, virtually infinite. The formulas describing the calculation of
parameters are not practical and are seldom used. They are provided here primarily to
teach the concept and the notation. In Chapter 7, we introduce probability distributions, which describe populations. At that time we show how parameters are calculated
from probability distributions. In general, small data sets of the type we feature in this
book are samples.
4 . 1 MEASURES
OF
C E N T R A L L O C AT I O N
Arithmetic Mean
There are three different measures that we use to describe the center of a set of data.
The first is the best known, the arithmetic mean, which we’ll refer to simply as the
mean. Students may be more familiar with its other name, the average. The mean is
computed by summing the observations and dividing by the number of observations.
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NUMERICAL DESCRIPTIVE TECHNIQUES
99
We label the observations in a sample x1, x2, Á , xn, where x1 is the first observation, x2
is the second, and so on until xn, where n is the sample size. As a result, the sample mean
is denoted x. In a population, the number of observations is labeled N and the population mean is denoted by m (Greek letter mu).
Mean
N
a xi
Population mean: m =
n
i=1
N
a xi
Sample mean: x =
EXAMPLE
4.1
i=1
n
Mean Time Spent on the Internet
A sample of 10 adults was asked to report the number of hours they spent on the
Internet the previous month. The results are listed here. Manually calculate the sample
mean.
0
7
12
5
33
14
8
0
9
22
SOLUTION
Using our notation, we have x1 = 0, x2 = 7, . . . , x10 = 22, and n = 10. The sample
mean is
n
a xi
x =
EXAMPLE
4.2
DATA
Xm03-01
i=1
n
=
0 + 7 + 12 + 5 + 33 + 14 + 8 + 0 + 9 + 22
110
=
= 11.0
10
10
Mean Long-Distance Telephone Bill
Refer to Example 3.1. Find the mean long-distance telephone bill.
SOLUTION
To calculate the mean, we add the observations and divide the sum by the size of the
sample. Thus,
n
a xi
x =
i=1
n
=
42.19 + 38.45 + Á + 45.77
8717.52
=
= 43.59
200
200
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Using the Computer
There are several ways to command Excel and Minitab to compute the mean. If we simply want to compute the mean and no other statistics, we can proceed as follows.
EXCEL
INSTRUCTIONS
Type or import the data into one or more columns. (Open Xm03-01.) Type into any
empty cell
⫽AVERAGE([Input range])
For Example 4.2, we would type into any cell
⫽AVERAGE(A1:A201)
The active cell would store the mean as 43.5876.
MINITAB
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-01.)
2. Click Calc and Column Statistics . . . . Specify Mean in the Statistic box. Type or
use the Select button to specify the Input variable and click OK. The sample mean
is outputted in the session window as 43.5876.
Median
The second most popular measure of central location is the median.
Median
The median is calculated by placing all the observations in order (ascending
or descending). The observation that falls in the middle is the median. The
sample and population medians are computed in the same way.
When there is an even number of observations, the median is determined by averaging the two observations in the middle.
EXAMPLE
4.3
Median Time Spent on Internet
Find the median for the data in Example 4.1.
SOLUTION
When placed in ascending order, the data appear as follows:
0
0
5
7
8
9
12
14
22
33
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101
The median is the average of the fifth and sixth observations (the middle two), which
are 8 and 9, respectively. Thus, the median is 8.5.
EXAMPLE
4.4
DATA
Xm03-01
Median Long-Distance Telephone Bill
Find the median of the 200 observations in Example 3.1.
SOLUTION
All the observations were placed in order. We observed that the 100th and 101st observations are 26.84 and 26.97, respectively. Thus, the median is the average of these two
numbers:
Median =
26.84 + 26.97
= 26.905
2
EXCEL
INSTRUCTIONS
To calculate the median, substitute MEDIAN in place of AVERAGE in the instructions
for the mean (page 100). The median is reported as 26.905.
MINITAB
INSTRUCTIONS
Follow the instructions for the mean to compute the mean except click Median instead
of Mean. The median is outputted as 26.905 in the session window.
INTERPRET
Half the observations are below 26.905, and half the observations are above 26.905.
Mode
The third and last measure of central location that we present here is the mode.
Mode
The mode is defined as the observation (or observations) that occurs with
the greatest frequency. Both the statistic and parameter are computed in the
same way.
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For populations and large samples, it is preferable to report the modal class, which
we defined in Chapter 2.
There are several problems with using the mode as a measure of central location.
First, in a small sample it may not be a very good measure. Second, it may not be
unique.
EXAMPLE
4.5
Mode Time Spent on Internet
Find the mode for the data in Example 4.1.
SOLUTION
All observations except 0 occur once. There are two 0s. Thus, the mode is 0. As you can
see, this is a poor measure of central location. It is nowhere near the center of the data.
Compare this with the mean 11.0 and median 8.5 and you can appreciate that in this
example the mean and median are superior measures.
EXAMPLE
4.6
DATA
Xm03-01
Mode of Long-Distance Bill
Determine the mode for Example 3.1.
SOLUTION
An examination of the 200 observations reveals that, except for 0, it appears that each
number is unique. However, there are 8 zeroes, which indicates that the mode is 0.
EXCEL
INSTRUCTIONS
To compute the mode, substitute MODE in place of AVERAGE in the previous instructions. Note that if there is more than one mode, Excel prints only the smallest one,
without indicating whether there are other modes. In this example, Excel reports that the
mode is 0.
MINITAB
Follow the instructions to compute the mean except click Mode instead of Mean. The
mode is outputted as 0 in the session window. (See page 20.)
Excel and Minitab: Printing All the Measures of Central Location plus
Other Statistics Both Excel and Minitab can produce the measures of central location plus a variety of others that we will introduce in later sections.
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103
EXCEL
Excel Output for Examples 4.2, 4.4, and 4.6
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B
Bills
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
43.59
2.76
26.91
0
38.97
1518.64
-1.29
0.54
119.63
0
119.63
8717.5
200
Excel reports the mean, median, and mode as the same values we obtained previously.
Most of the other statistics will be discussed later.
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-01.)
2. Click Data, Data Analysis, and Descriptive Statistics.
3. Specify the Input Range (A1:A201) and click Summary statistics.
MINITAB
Minitab Output for Examples 4.2, 4.4, and 4.6
Descriptive Statistics: Bills
Variable
Bills
Mean
43.59
Median
26.91
Mode
0
N for
Mode
8
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm03-01.)
2. Click Stat, Basic Statistics, and Display Descriptive Statistics . . . .
3. Type or use Select to identify the name of the variable or column (Bills). Click
Statistics . . . to add or delete particular statistics.
Mean, Median, Mode: Which Is Best?
With three measures from which to choose, which one should we use? There are several factors to consider when making our choice of measure of central location. The
mean is generally our first selection. However, there are several circumstances when
the median is better. The mode is seldom the best measure of central location. One
advantage the median holds is that it is not as sensitive to extreme values as is the mean.
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To illustrate, consider the data in Example 4.1. The mean was 11.0, and the median
was 8.5. Now suppose that the respondent who reported 33 hours actually reported 133
hours (obviously an Internet addict). The mean becomes
n
a xi
x =
i=1
=
n
0 + 7 + 12 + 5 + 133 + 14 + 8 + 0 + 22
210
=
= 21.0
10
10
This value is exceeded by only 2 of the 10 observations in the sample, making this
statistic a poor measure of central location. The median stays the same. When there is a
relatively small number of extreme observations (either very small or very large, but not
both), the median usually produces a better measure of the center of the data.
To see another advantage of the median over the mean, suppose you and your
classmates have written a statistics test and the instructor is returning the graded tests.
What piece of information is most important to you? The answer, of course, is your
mark. What is the next important bit of information? The answer is how well you performed relative to the class. Most students ask their instructor for the class mean. This
is the wrong statistic to request. You want the median because it divides the class into
two halves. This information allows you to identify which half of the class your mark
falls into. The median provides this information; the mean does not. Nevertheless, the
mean can also be useful in this scenario. If there are several sections of the course, the
section means can be compared to determine whose class performed best (or worst).
Measures of Central Location for Ordinal and Nominal Data
When the data are interval, we can use any of the three measures of central location.
However, for ordinal and nominal data, the calculation of the mean is not valid. Because
the calculation of the median begins by placing the data in order, this statistic is appropriate for ordinal data. The mode, which is determined by counting the frequency of
each observation, is appropriate for nominal data. However, nominal data do not have a
“center,” so we cannot interpret the mode of nominal data in that way. It is generally
pointless to compute the mode of nominal data.
APPLICATIONS in FINANCE
Geometric Mean
© Image State Royalty-free
The arithmetic mean is the single most popular and useful measure of central
location. We noted certain situations where the median is a better measure of
central location. However, there is another circumstance where neither the
mean nor the median is the best measure. When the variable is a growth rate or
rate of change, such as the value of an investment over periods of time, we need
another measure. This will become apparent from the following illustration.
Suppose you make a 2-year investment of $1,000, and it grows by 100% to
$2,000 during the first year. During the second year, however, the investment suffers a
50% loss, from $2,000 back to $1,000. The rates of return for years 1 and 2 are R1 ⫽ 100% and
R2 ⫽ ⫺50%, respectively. The arithmetic mean (and the median) is computed as
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NUMERICAL DESCRIPTIVE TECHNIQUES
R =
R1 + R2
2
=
105
100 + (-50)
= 25%
2
But this figure is misleading. Because there was no change in the value of the investment from
the beginning to the end of the 2-year period, the “average” compounded rate of return is 0%.
As you will see, this is the value of the geometric mean.
Let Ri denote the rate of return (in decimal form) in period i (i ⫽ 1, 2, . . . , n). The geometric
mean Rg of the returns R1,R2, Á ,Rn is defined such that
(1 + Rg)n = (1 + R1)(1 + R2) Á (1 + Rn)
Solving for Rg, we produce the following formula:
n
Rg = 2(1 + R1)(1 + R2) Á (1 + Rn) - 1
The geometric mean of our investment illustration is
Rg = 2(1 + R1)(1 + R2) Á (1 + Rn) - 1 = 2 (1 + 1)(1 + 3 -.504) - 1 = 1 - 1 = 0
n
2
The geometric mean is therefore 0%. This is the single “average” return that allows us to compute
the value of the investment at the end of the investment period from the beginning value. Thus,
using the formula for compound interest with the rate ⫽ 0%, we find
Value at the end of the investment period ⫽ 1,000(1 ⫹ Rg)2 ⫽ 1,000(1 ⫹ 0) 2 ⫽ 1,000
The geometric mean is used whenever we wish to find the “average” growth rate, or rate of
change, in a variable over time. However, the arithmetic mean of n returns (or growth rates) is the
appropriate mean to calculate if you wish to estimate the mean rate of return (or growth rate) for
any single period in the future; that is, in the illustration above if we wanted to estimate the rate
of return in year 3, we would use the arithmetic mean of the two annual rates of return, which
we found to be 25%.
EXCEL
INSTRUCTIONS
1. Type or import the values of 1 ⫹ Ri into a column.
2. Follow the instructions to produce the mean (page 100) except substitute
GEOMEAN in place of AVERAGE.
3. To determine the geometric mean, subtract 1 from the number produced.
MINITAB
Minitab does not compute the geometric mean.
Here is a summary of the numerical techniques introduced in this section and when to
use them.
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Factors That Identify When to Compute the Mean
1. Objective: Describe a single set of data
2. Type of data: Interval
3. Descriptive measurement: Central location
Factors That Identify When to Compute the Median
1. Objective: Describe a single set of data
2. Type of data: Ordinal or interval (with extreme observations)
3. Descriptive measurement: Central location
Factors That Identify When to Compute the Mode
1. Objective: Describe a single set of data
2. Type of data: Nominal, ordinal, interval
Factors That Identify When to Compute the Geometric Mean
1. Objective: Describe a single set of data
2. Type of data: Interval; growth rates
EXERCISES
4.1 A sample of 12 people was asked how much change
they had in their pockets and wallets. The responses
(in cents) are
52
60
25
30
15
33
0
81
104
40
4.4 The midterm test for a statistics course has a time
44
5
Determine the mean, median, and mode for these
data.
4.2 The number of sick days due to colds and flu last year
was recorded by a sample of 15 adults. The data are
5
3
7
8
0
10
3
5
15
2
6
0
5
12
9
Compute the mean, median, and mode.
4.3 A random sample of 12 joggers was asked to keep
track and report the number of miles they ran last
week. The responses are
5.5
5.3
7.2
3.4
1.6
12.5
22.0
18.6
8.7
8.3
a. Compute the three statistics that measure central
location.
b. Briefly describe what each statistic tells you.
2.8
6.6
limit of 1 hour. However, like most statistics exams
this one was quite easy. To assess how easy, the
professor recorded the amount of time taken by a
sample of nine students to hand in their test
papers. The times (rounded to the nearest minute)
are
33
29
45
60
42
19
52
38
36
a. Compute the mean, median, and mode.
b. What have you learned from the three statistics
calculated in part (a)?
4.5 The professors at Wilfrid Laurier University are
required to submit their final exams to the registrar’s office 10 days before the end of the semester.
The exam coordinator sampled 20 professors and
recorded the number of days before the final exam
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NUMERICAL DESCRIPTIVE TECHNIQUES
that each submitted his or her exam. The results
are
14
7
8
4
3 2 6
9 13 15
4 9 13 10 12
8 11 12 4 0
a. Compute the mean, median, and mode.
b. Briefly describe what each statistic tells you.
4.6 Compute the geometric mean of the following rates
of return.
.25 ⫺.10
.50
4.7 What is the geometric mean of the following rates
of return?
.50
.30 ⫺.50
ment over a 5-year period.
1
.10
2
.22
3
.06
4
⫺.05
5
.20
a. Compute the mean and median of the returns.
b. Compute the geometric mean.
c. Which one of the three statistics computed in
parts (a) and (b) best describes the return over the
5-year period? Explain.
4.9 An investment you made 5 years ago has realized the
following rates of return.
Year
Rate of Return
1
2
⫺.15 ⫺.20
3
.15
4
5
⫺.08 .50
a. Compute the mean and median of the rates of
return.
b. Compute the geometric mean.
c. Which one of the three statistics computed in
parts (a) and (b) best describes the return over the
5-year period? Explain.
4.10 An investment of $1,000 you made 4 years ago was
worth $1,200 after the first year, $1,200 after the second year, $1,500 after the third year, and $2,000 today.
a. Compute the annual rates of return.
b. Compute the mean and median of the rates of
return.
c. Compute the geometric mean.
d. Discuss whether the mean, median, or geometric
mean is the best measure of the performance of
the investment.
4.11 Suppose that you bought a stock 6 years ago at
$12.The stock’s price at the end of each year is
shown here.
Year
Price
1
10
2
14
c. Compute the geometric mean of the rates of
return.
d. Explain why the best statistic to use to describe
what happened to the price of the stock over the
6-year period is the geometric mean.
The following exercises require the use of a computer and
software.
4.12 Xr04-12 The starting salaries of a sample of 125
recent MBA graduates are recorded.
a. Determine the mean and median of these data.
b. What do these two statistics tell you about the
starting salaries of MBA graduates?
4.13 Xr04-13 To determine whether changing the color of
⫺.25
4.8 The following returns were realized on an investYear
Rate of Return
107
3
15
4
22
5
30
6
25
a. Compute the rate of return for each year.
b. Compute the mean and median of the rates of
return.
its invoices would improve the speed of payment, a
company selected 200 customers at random and sent
their invoices on blue paper. The number of days
until the bills were paid was recorded. Calculate the
mean and median of these data. Report what you
have discovered.
4.14 Xr04-14 A survey undertaken by the U.S. Bureau of
Labor Statistics, Annual Consumer Expenditure,
asks American adults to report the amount of money
spent on reading material in 2006. (Source: Adapted
from Statistical Abstract of the United States, 2009,
Table 664.)
a. Compute the mean and median of the sample.
b. What do the statistics computed in part (a) tell
you about the reading materials expenditures?
4.15 Xr04-15 A survey of 225 workers in Los Angeles and
190 workers in New York asked each to report the
average amount of time spent commuting to work.
(Source: Adapted from Statistical Abstract of the United
States, 2009, Table 1060.)
a. Compute the mean and median of the commuting times for workers in Los Angeles.
b. Repeat part (a) for New York workers.
c. Summarize your findings.
4.16 Xr04-16 In the United States, banks and financial
institutions often require buyers of houses to pay
fees in order to arrange mortgages. In a survey conducted by the U.S. Federal Housing Finance Board,
350 buyers of new houses who received a mortgage
from a bank were asked to report the amount of fees
(fees include commissions, discounts, and points)
they paid as a percentage of the whole mortgage.
(Source: Adapted from Statistical Abstract of the United
States, 2009, Table 1153.)
a. Compute the mean and median.
b. Interpret the statistics you computed.
4.17 Xr04-17 In an effort to slow drivers, traffic engineers
painted a solid line 3 feet from the curb over the entire
length of a road and filled the space with diagonal
lines. The lines made the road look narrower. A sample of car speeds was taken after the lines were drawn.
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a. Compute the mean, median, and mode of these
data.
b. Briefly describe the information you acquired
from each statistic calculated in part (a).
surveyed and asked to report the amount of money
spent annually on fruits and vegetables. Compute
the mean and median of these data and interpret the
results. (Source: Adapted from Statistical Abstract of
the United States, 2009, Table 662.)
4.18 Xr04-18 How much do Americans spend on various
food groups? Two hundred American families were
4 . 2 MEASURES
OF
VA R I A B I L I T Y
The statistics introduced in Section 4.1 serve to provide information about the central
location of the data. However, as we have already discussed in Chapter 2, there are
other characteristics of data that are of interest to practitioners of statistics. One such
characteristic is the spread or variability of the data. In this section, we introduce four
measures of variability. We begin with the simplest.
Range
Range
Range ⫽ Largest observation ⫺ Smallest observation
The advantage of the range is its simplicity. The disadvantage is also its simplicity.
Because the range is calculated from only two observations, it tells us nothing about the
other observations. Consider the following two sets of data.
Set 1:
Set 2:
4
4
4
8
4
15
4
24
4
39
50
50
The range of both sets is 46. The two sets of data are completely different, yet their
ranges are the same. To measure variability, we need other statistics that incorporate all
the data and not just two observations.
Variance
The variance and its related measure, the standard deviation, are arguably the most
important statistics. They are used to measure variability, but, as you will discover, they
play a vital role in almost all statistical inference procedures.
Variance
N
2
a (xi - m)
Population variance:
i=1
s2 =
N
n
2
a (xi - x)
Sample variance:*
s2 =
i=1
n - 1
The population variance is represented by s2(Greek letter sigma squared).
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109
Examine the formula for the sample variance s2. It may appear to be illogical that
in calculating s2 we divide by n ⫺ 1 rather than by n.* However, we do so for the following reason. Population parameters in practical settings are seldom known. One
objective of statistical inference is to estimate the parameter from the statistic. For
example, we estimate the population mean m from the sample mean x. Although it is
not obviously logical, the statistic created by dividing a (xi - x)2 by n ⫺ 1 is a better
estimator than the one created by dividing by n. We will discuss this issue in greater
detail in Section 10.1.
To compute the sample variance s2, we begin by calculating the sample mean x.
Next we compute the difference (also call the deviation) between each observation and
the mean. We square the deviations and sum. Finally, we divide the sum of squared
deviations by n ⫺ 1.
We’ll illustrate with a simple example. Suppose that we have the following observations of the numbers of hours five students spent studying statistics last week:
8
4
9
11
3
The mean is
x =
35
8 + 4 + 9 + 11 + 3
=
= 7
5
5
For each observation, we determine its deviation from the mean. The deviation is
squared, and the sum of squares is determined as shown in Table 4.1.
TABLE
4 .1
Calculation of Sample Variance
(xi - x)2
xi
(xi - x)
8
(8 ⫺ 7) ⫽ 1
4
(4 ⫺ 7) ⫽ –3
9
(9 ⫺ 7) ⫽ 2
(2)2 ⫽ 4
11
(11 ⫺ 7) ⫽ 4
(4)2 ⫽ 16
(1)2 ⫽ 1
(⫺3)2 ⫽ 9
(3 ⫺ 7) ⫽ ⫺4
3
5
a (xi - x) = 0
I=1
(⫺4)2 ⫽ 16
5
2
a (xi - x) = 46
I=1
The sample variance is
n
2
a (xi - x)
s2 =
i=1
n - 1
=
46
= 11.5
5 - 1
The calculation of this statistic raises several questions. Why do we square the deviations before averaging? If you examine the deviations, you will see that some of the
*Technically, the variance of the sample is calculated by dividing the sum of squared deviations by n. The
statistic computed by dividing the sum of squared deviations by n ⫺ 1 is called the sample variance corrected
for the mean. Because this statistic is used extensively, we will shorten its name to sample variance.
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deviations are positive and some are negative. When you add them together, the sum is 0.
This will always be the case because the sum of the positive deviations will always equal
the sum of the negative deviations. Consequently, we square the deviations to avoid the
“canceling effect.”
Is it possible to avoid the canceling effect without squaring? We could average the
absolute value of the deviations. In fact, such a statistic has already been invented. It is
called the mean absolute deviation or MAD. However, this statistic has limited utility
and is seldom calculated.
What is the unit of measurement of the variance? Because we squared the deviations, we also squared the units. In this illustration the units were hours (of study).
Thus, the sample variance is 11.5 hours2.
EXAMPLE
4 .7
Summer Jobs
The following are the number of summer jobs a sample of six students applied for. Find
the mean and variance of these data.
17
15
23
7
9
13
SOLUTION
The mean of the six observations is
x =
84
17 + 15 + 23 + 7 + 9 + 13
=
= 14 jobs
6
6
The sample variance is
n
2
a (xi - x)
s2 =
i=1
n - 1
(17 - 14)2 + (15 - 14)2 + (23 - 14)2 + (7 - 14)2 + (9 - 14)2 + (13 - 14)2
=
=
6 - 1
9 + 1 + 81 + 49 + 25 + 1
166
=
= 33.2 jobs2
5
5
(Optional) Shortcut Method for Variance The calculations for larger data sets
are quite time-consuming. The following shortcut for the sample variance may help
lighten the load.
Shortcut for Sample Variance
n
s2 =
n
1
E a x2i n - 1 i=1
¢ a xi ≤
i=1
n
2
U
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111
To illustrate, we’ll do Example 4.7 again.
n
2
2
2
2
2
2
2
a xi = 17 + 15 + 23 + 7 + 9 + 13 = 1,342
I=1
n
a xi = 17 + 15 + 23 + 7 + 9 + 13 = 84
i=1
n
2
¢ a xi ≤ = 842 = 7,056
i=1
n
s2 =
n
1
E a x2i n - 1 i=1
¢ a xi ≤
i=1
n
2
U =
7056
1
c1342 d = 33.2 jobs2
6 - 1
6
Notice that we produced the same exact answer.
EXCEL
INSTRUCTIONS
Follow the instructions to compute the mean (page 100) except type VAR instead of
AVERAGE.
MINITAB
1. Type or import data into one column.
2. Click Stat, Basic Statistics, Display Descriptive Statistics . . . , and select the
variable.
3. Click Statistics and Variance.
Interpreting the Variance
We calculated the variance in Example 4.7 to be 33.2 jobs2. What does this statistic tell
us? Unfortunately, the variance provides us with only a rough idea about the amount of
variation in the data. However, this statistic is useful when comparing two or more sets
of data of the same type of variable. If the variance of one data set is larger than that of
a second data set, we interpret that to mean that the observations in the first set display
more variation than the observations in the second set.
The problem of interpretation is caused by the way the variance is computed.
Because we squared the deviations from the mean, the unit attached to the variance is the
square of the unit attached to the original observations. In other words, in Example 4.7
the unit of the data is jobs; the unit of the variance is jobs squared. This contributes to
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the problem of interpretation. We resolve this difficulty by calculating another related
measure of variability.
Standard Deviation
Standard Deviation
Population standard deviation: s
Sample standard deviation: s
= 2s2
= 2s2
The standard deviation is simply the positive square root of the variance. Thus, in
Example 4.7, the sample standard deviation is
s = 2s2 = 233.2 = 5.76 jobs
Notice that the unit associated with the standard deviation is the unit of the original
data set.
EXAMPLE
4.8
DATA
Xm04-08
Comparing the Consistency of Two Types of Golf Clubs
Consistency is the hallmark of a good golfer. Golf equipment manufacturers are constantly
seeking ways to improve their products. Suppose that a recent innovation is designed to
improve the consistency of its users. As a test, a golfer was asked to hit 150 shots using a
7 iron, 75 of which were hit with his current club and 75 with the new innovative 7 iron.
The distances were measured and recorded. Which 7 iron is more consistent?
SOLUTION
To gauge the consistency, we must determine the standard deviations. (We could also compute the variances, but as we just pointed out, the standard deviation is easier to interpret.)
We can get Excel and Minitab to print the sample standard deviations. Alternatively, we
can calculate all the descriptive statistics, a course of action we recommend because we
often need several statistics. The printouts for both 7 irons are shown here.
EXCEL
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B
Current
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
150.55
0.67
151
150
5.79
33.55
0.13
-0.43
28
134
162
11291
75
C
D
Innovation
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
E
150.15
0.36
150
149
3.09
9.56
-0.89
0.18
12
144
156
11261
75
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113
MINITAB
Descriptive Statistics: Current, Innovation
Variable
N
Current
75
Innovation 75
N* Mean
0 150.55
0 150.15
StDev
5.79
3.09
Variance
33.55
9.56
Minimum
Q1
134.00 148.00
144.00 148.00
Median
Q3
151.00 155.00
150.00 152.00
Variable
Maximum
Current
162.00
Innovation
156.00
INTERPRET
The standard deviation of the distances of the current 7 iron is 5.79 yards whereas that
of the innovative 7 iron is 3.09 yards. Based on this sample, the innovative club is more
consistent. Because the mean distances are similar it would appear that the new club is
indeed superior.
Interpreting the Standard Deviation Knowing the mean and standard deviation
allows the statistics practitioner to extract useful bits of information. The information
depends on the shape of the histogram. If the histogram is bell shaped, we can use the
Empirical Rule.
Empirical Rule
1. Approximately 68% of all observations fall within one standard
deviation of the mean.
2. Approximately 95% of all observations fall within two standard
deviations of the mean.
3. Approximately 99.7% of all observations fall within three standard
deviations of the mean.
EXAMPLE
4 .9
Using the Empirical Rule to Interpret Standard
Deviation
After an analysis of the returns on an investment, a statistics practitioner discovered that
the histogram is bell shaped and that the mean and standard deviation are 10% and 8%,
respectively. What can you say about the way the returns are distributed?
SOLUTION
Because the histogram is bell shaped, we can apply the Empirical Rule:
1. Approximately 68% of the returns lie between 2% (the mean minus one standard
deviation ⫽ 10 ⫺ 8) and 18% (the mean plus one standard deviation ⫽ 10 ⫹ 8).
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2. Approximately 95% of the returns lie between ⫺6% [the mean minus two standard deviations ⫽ 10 ⫺ 2(8)] and 26% [the mean plus two standard deviations
⫽ 10 ⫹ 2(8)].
3. Approximately 99.7% of the returns lie between ⫺14% [the mean minus three
standard deviations ⫽ 10 ⫺ 3(8)] and 34% [the mean plus three standard deviations ⫽ 10 ⫺ 3(8)].
A more general interpretation of the standard deviation is derived from Chebysheff’s
Theorem, which applies to all shapes of histograms.
Chebysheff’s Theorem
The proportion of observations in any sample or population that lie within
k standard deviations of the mean is at least
1 -
1
for k 7 1
k2
When k ⫽ 2, Chebysheff’s Theorem states that at least three-quarters (75%) of
all observations lie within two standard deviations of the mean. With k ⫽ 3,
Chebysheff’s Theorem states that at least eight-ninths (88.9%) of all observations lie
within three standard deviations of the mean.
Note that the Empirical Rule provides approximate proportions, whereas
Chebysheff’s Theorem provides lower bounds on the proportions contained in the
intervals.
EXAMPLE
4.10
Using Chebysheff’s Theorem to Interpret Standard
Deviation
The annual salaries of the employees of a chain of computer stores produced a positively skewed histogram. The mean and standard deviation are $28,000 and $3,000,
respectively. What can you say about the salaries at this chain?
SOLUTION
Because the histogram is not bell shaped, we cannot use the Empirical Rule. We must
employ Chebysheff’s Theorem instead.
The intervals created by adding and subtracting two and three standard deviations
to and from the mean are as follows:
1. At least 75% of the salaries lie between $22,000 [the mean minus two standard
deviations ⫽ 28,000 ⫺ 2(3,000)] and $34,000 [the mean plus two standard deviations ⫽ 28,000 ⫹ 2(3,000)].
2. At least 88.9% of the salaries lie between $19,000 [the mean minus three standard
deviations ⫽ 28,000 ⫺ 3(3,000)] and $37,000 [the mean plus three standard deviations ⫽ 28,000 ⫹ 3(3,000)].
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115
Coefficient of Variation
Is a standard deviation of 10 a large number indicating great variability or a small number
indicating little variability? The answer depends somewhat on the magnitude of the observations in the data set. If the observations are in the millions, then a standard deviation of
10 will probably be considered a small number. On the other hand, if the observations are
less than 50, then the standard deviation of 10 would be seen as a large number. This logic
lies behind yet another measure of variability, the coefficient of variation.
Coefficient of Variation
The coefficient of variation of a set of observations is the standard deviation
of the observations divided by their mean:
Population coefficient of variation: CV =
Sample coefficient of variation: cv =
s
m
s
x
Measures of Variability for Ordinal and Nominal Data
The measures of variability introduced in this section can be used only for interval data.
The next section will feature a measure that can be used to describe the variability of
ordinal data. There are no measures of variability for nominal data.
Approximating the Mean and Variance from Grouped Data
The statistical methods presented in this chapter are used to compute descriptive
statistics from data. However, in some circumstances, the statistics practitioner does
not have the raw data but instead has a frequency distribution. This is often the case
when data are supplied by government organizations. In Appendix Approximating
Means and Variances for Grouped Data on Keller’s website we provide the formulas
used to approximate the sample mean and variance.
We complete this section by reviewing the factors that identify the use of measures
of variability.
Factors That Identify When to Compute the Range, Variance, Standard
Deviation, and Coefficient of Variation
1. Objective: Describe a single set of data
2. Type of Data: Interval
3. Descriptive measurement: Variability
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CHAPTER 4
EXERCISES
4.19 Calculate the variance of the following data.
9
3
7
4
1
7
5
respectively. What can you say about the proportions of observations that lie between each of the
following intervals?
a. 90 and 150
b. 60 and 180
c. 30 and 210
4
4.20 Calculate the variance of the following data.
4
5
3
6
5
6
5
6
4.21 Determine the variance and standard deviation of
the following sample.
12
6
22
4.31 Xr04-31 There has been much media coverage of the
31 23 13 15 17 21
4.22 Find the variance and standard deviation of the fol-
lowing sample.
0
⫺5
⫺3
6
The following exercises require a computer and software.
4
⫺4
1 ⫺5
0
3
4.23 Examine the three samples listed here. Without per-
forming any calculations, indicate which sample has
the largest amount of variation and which sample
has the smallest amount of variation. Explain how
you produced your answer.
a. 17
29
12
16
11
b. 22
18
23
20
17
c. 24
37
6
39
29
high cost of medicinal drugs in the United States.
One concern is the large variation from pharmacy to
pharmacy. To investigate, a consumer advocacy
group took a random sample of 100 pharmacies
around the country and recorded the price (in dollars per 100 pills) of Prozac. Compute the range,
variance, and standard deviation of the prices.
Discuss what these statistics tell you.
4.32 Xr04-32 Many traffic experts argue that the most
each part. Was your answer in Exercise 4.23 correct?
important factor in accidents is not the average
speed of cars but the amount of variation. Suppose
that the speeds of a sample of 200 cars were taken
over a stretch of highway that has seen numerous
accidents. Compute the variance and standard deviation of the speeds, and interpret the results.
4.25 A friend calculates a variance and reports that it is
4.33 Xr04-33 Three men were trying to make the football
4.24 Refer to Exercise 4.23. Calculate the variance for
–25.0. How do you know that he has made a serious
calculation error?
4.26 Create a sample of five numbers whose mean is
6 and whose standard deviation is 0.
4.27 A set of data whose histogram is bell shaped yields a
mean and standard deviation of 50 and 4, respectively.
Approximately what proportion of observations
a. are between 46 and 54?
b. are between 42 and 58?
c. are between 38 and 62?
4.28 Refer to Exercise 4.27. Approximately what propor-
tion of observations
a. are less than 46?
b. are less than 58?
c. are greater than 54?
4.29 A set of data whose histogram is extremely skewed
yields a mean and standard deviation of 70 and 12,
respectively. What is the minimum proportion of
observations that
a. are between 46 and 94?
b. are between 34 and 106?
4.30 A statistics practitioner determined that the mean
and standard deviation of a data set were 120 and 30,
team as punters. The coach had each of them punt
the ball 50 times, and the distances were recorded.
a. Compute the variance and standard deviation for
each punter.
b. What do these statistics tell you about the punters?
4.34 Xr04-34 Variance is often used to measure quality in
production-line products. Suppose that a sample of
steel rods that are supposed to be exactly 100 cm
long is taken. The length of each is determined, and
the results are recorded. Calculate the variance and
the standard deviation. Briefly describe what these
statistics tell you.
4.35 Xr04-35 To learn more about the size of withdrawals
at a banking machine, the proprietor took a sample
of 75 withdrawals and recorded the amounts.
Determine the mean and standard deviation of these
data, and describe what these two statistics tell you
about the withdrawal amounts.
4.36 Xr04-36 Everyone is familiar with waiting lines or
queues. For example, people wait in line at a supermarket to go through the checkout counter. There
are two factors that determine how long the queue
becomes. One is the speed of service. The other is
the number of arrivals at the checkout counter. The
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117
a. Compute the standard deviation of the number
of arrivals.
b. Assuming that the histogram is bell shaped,
interpret the standard deviation.
mean number of arrivals is an important number,
but so is the standard deviation. Suppose that a consultant for the supermarket counts the number of
arrivals per hour during a sample of 150 hours.
A M E R I C A N N AT I O N A L E L E C T I O N S U R V E Y E X E R C I S E S
4.37 ANES2008* The ANES in 2008 asked respondents to
4.38 ANES2008* Respondents were asked to report the
state their ages stored as AGE.
a. Calculate the mean, variance, and standard deviation.
b. Draw a histogram.
c. Use the Empirical Rule, if applicable, or
Chebysheff’s Theorem to interpret the mean and
standard deviation.
number of minutes spent watching news on television during a typical day (TIME2).
a. Calculate the mean and standard deviation.
b. Draw a histogram.
c. Use the Empirical Rule, if applicable, or
Chebysheff’s Theorem to interpret the mean and
standard deviation.
GENER AL SO CIAL SURVEY EXERCISE
4.39 GSS2008* One of the questions in the 2008 General
Social Survey was, If you were born outside the
United States, at what age did you permanently
move to the United States (AGECMEUS)?
a. Calculate the mean, variance, and standard
deviation.
4 . 3 MEASURES
OF
R E L AT I V E S TA N D I N G
b. Draw a histogram.
c. Use the Empirical Rule, if applicable, or
Chebysheff’s Theorem to interpret the mean and
standard deviation.
AND
B OX P LOT S
Measures of relative standing are designed to provide information about the position of
particular values relative to the entire data set. We’ve already presented one measure of
relative standing, the median, which is also a measure of central location. Recall that the
median divides the data set into halves, allowing the statistics practitioner to determine
which half of the data set each observation lies in. The statistics we’re about to introduce will give you much more detailed information.
Percentile
The Pth percentile is the value for which P percent are less than that value
and (100–P)% are greater than that value.
The scores and the percentiles of the Scholastic Achievement Test (SAT) and the
Graduate Management Admission Test (GMAT), as well as various other admissions
tests, are reported to students taking them. Suppose for example, that your SAT score is
reported to be at the 60th percentile. This means that 60% of all the other marks are
below yours and 40% are above it. You now know exactly where you stand relative to
the population of SAT scores.
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CHAPTER 4
We have special names for the 25th, 50th, and 75th percentiles. Because these
three statistics divide the set of data into quarters, these measures of relative standing
are also called quartiles. The first or lower quartile is labeled Q1. It is equal to the 25th
percentile. The second quartile, Q2, is equal to the 50th percentile, which is also the
median. The third or upper quartile, Q3, is equal to the 75th percentile. Incidentally,
many people confuse the terms quartile and quarter. A common error is to state that
someone is in the lower quartile of a group when they actually mean that someone is in
the lower quarter of a group.
Besides quartiles, we can also convert percentiles into quintiles and deciles.
Quintiles divide the data into fifths, and deciles divide the data into tenths.
Locating Percentiles
The following formula allows us to approximate the location of any percentile.
Location of a Percentile
LP = (n + 1)
P
100
where LP is the location of the Pth percentile.
EXAMPLE
4.11
Percentiles of Time Spent on Internet
Calculate the 25th, 50th, and 75th percentiles (first, second, and third quartiles) of the
data in Example 4.1.
SOLUTION
Placing the 10 observations in ascending order we get
0
0
5
7
8
9
12
14
22
33
The location of the 25th percentile is
L25 = (10 + 1)
25
= (11)(.25) = 2.75
100
The 25th percentile is three-quarters of the distance between the second (which is 0)
and the third (which is 5) observations. Three-quarters of the distance is
(.75)(5 ⫺ 0) ⫽ 3.75
Because the second observation is 0, the 25th percentile is 0 ⫹ 3.75 ⫽ 3.75.
To locate the 50th percentile, we substitute P ⫽ 50 into the formula and produce
L50 = (10 + 1)
50
= (11)(.5) = 5.5
100
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NUMERICAL DESCRIPTIVE TECHNIQUES
119
which means that the 50th percentile is halfway between the fifth and sixth observations.
The fifth and sixth observations are 8 and 9, respectively. The 50th percentile is 8.5. This
is the median calculated in Example 4.3.
The 75th percentile’s location is
L75 = (10 + 1)
75
= (11)(.75) = 8.25
100
Thus, it is located one-quarter of the distance between the eighth and ninth observations, which are 14 and 22, respectively. One-quarter of the distance is
(.25)(22 ⫺ 14) ⫽ 2
which means that the 75th percentile is
14 ⫹ 2 ⫽ 16
EXAMPLE
4.12
DATA
Xm03-01
Quartiles of Long-Distance Telephone Bills
Determine the quartiles for Example 3.1.
SOLUTION
EXCEL
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
B
Bills
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(50)
Smallest(50)
43.59
2.76
26.91
0
38.97
1518.64
-1.29
0.54
119.63
0
119.63
8717.52
200
85
9.22
INSTRUCTIONS
Follow the instructions for Descriptive Statistics (page 103). In the dialog box, click
Kth Largest and type in the integer closest to n/4. Repeat for Kth Smallest, typing in
the integer closest to n/4.
Excel approximates the third and first percentiles in the following way. The
Largest(50) is 85, which is the number such that 150 numbers are below it and 49 numbers are above it. The Smallest(50) is 9.22, which is the number such that 49 numbers
are below it and 150 numbers are above it. The median is 26.91, a statistic we discussed in
Example 4.4.
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MINITAB
Descriptive Statistics: Bills
Variable
Bills
Mean
43.59
StDev
38.97
Variance
1518.64
Minimum
0.00
Q1
9.28
Median
26.91
Q3
84.94
Maximum
119.63
Minitab outputs the first and third quartiles as Q1 (9.28) and Q3 (84.94), respectively.
(See page 103.)
We can often get an idea of the shape of the histogram from the quartiles. For example, if the first and second quartiles are closer to each other than are the second and third
quartiles, then the histogram is positively skewed. If the first and second quartiles are farther apart than the second and third quartiles, then the histogram is negatively skewed.
If the difference between the first and second quartiles is approximately equal to the difference between the second and third quartiles, then the histogram is approximately
symmetric. The box plot described subsequently is particularly useful in this regard.
Interquartile Range
The quartiles can be used to create another measure of variability, the interquartile
range, which is defined as follows.
Interquartile Range
Interquartile range ⫽ Q3 - Q1
The interquartile range measures the spread of the middle 50% of the observations. Large values of this statistic mean that the first and third quartiles are far apart,
indicating a high level of variability.
EXAMPLE
4.13
DATA
Xm03-01
Interquartile Range of Long-Distance Telephone Bills
Determine the interquartile range for Example 3.1.
SOLUTION
Using Excel’s approximations of the first and third quartiles, we find
Interquartile range = Q3 - Q1 = 85 - 9.22 = 75.78
Box Plots
Now that we have introduced quartiles we can present one more graphical technique,
the box plot. This technique graphs five statistics: the minimum and maximum observations, and the first, second, and third quartiles. It also depicts other features of a set of
data. Figure 4.1 exhibits the box plot of the data in Example 4.1.
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NUMERICAL DESCRIPTIVE TECHNIQUES
FIGURE
4.1
121
Box Plot for Example 4.1
0
5
10
15
20
25
30
35
The three vertical lines of the box are the first, second, and third quartiles. The lines
extending to the left and right are called whiskers. Any points that lie outside the whiskers
are called outliers. The whiskers extend outward to the smaller of 1.5 times the interquartile range or to the most extreme point that is not an outlier.
Outliers Outliers are unusually large or small observations. Because an outlier is considerably removed from the main body of the data set, its validity is suspect.
Consequently, outliers should be checked to determine that they are not the result of an
error in recording their values. Outliers can also represent unusual observations that
should be investigated. For example, if a salesperson’s performance is an outlier on the
high end of the distribution, the company could profit by determining what sets that
salesperson apart from the others.
EXAMPLE
4.14
DATA
Xm03-01
Box Plot of Long-Distance Telephone Bills
Draw the box plot for Example 3.1.
SOLUTION
EXCEL
0
20
40
60
80
100
120
INSTRUCTIONS
1. Type or import the data into one column or two or more adjacent columns. (Open
Xm03-01.)
2. Click Add-Ins, Data Analysis Plus, and Box Plot.
3. Specify the Input Range (A1:A201).
A box plot will be created for each column of data that you have specified or highlighted.
Notice that the quartiles produced in the Box Plot are not exactly the same as those
produced by Descriptive Statistics. The Box Plot command uses a slightly different
method than the Descriptive Methods command.
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MINITAB
Box Plot of Bills
0
20
40
60
80
100
120
Bills
INSTRUCTIONS
1. Type or import the data into one column or more columns. (Open Xm03-01.)
2. Click Graph and Box Plot . . . .
3. Click Simple if there is only one column of data or Multiple Y’s if there are two or
more columns.
4. Type or Select the variable or variables in the Graph variables box (Bills).
5. The box plot will be drawn so that the values will appear on the vertical axis. To turn
the box plot on its side click Scale . . . , Axes and Ticks, and Transpose value and
category scales.
INTERPRET
The smallest value is 0, and the largest is 119.63. The first, second, and third quartiles
are 9.275, 26.905, and 84.9425, respectively. The interquartile range is 75.6675. One
and one-half times the interquartile range is 1.5 * 75.6675 = 113.5013. Outliers are
defined as any observations that are less than 9.275 ⫺ 113.5013 ⫽ ⫺104.226 and any
observations that are larger than 84.9425 ⫹ 113.5013 ⫽ 198.4438. The whisker to the
left extends only to 0, which is the smallest observation that is not an outlier. The
whisker to the right extends to 119.63, which is the largest observation that is not an
outlier. There are no outliers.
The box plot is particularly useful when comparing two or more data sets.
EXAMPLE
4.15
DATA
Xm04-15
Comparing Service Times of Fast-Food Restaurants’
Drive-Throughs
A large number of fast-food restaurants with drive-through windows offer drivers and
their passengers the advantages of quick service. To measure how good the service is, an
organization called QSR planned a study in which the amount of time taken by a sample of drive-through customers at each of five restaurants was recorded. Compare the
five sets of data using a box plot and interpret the results.
SOLUTION
We use the computer and our software to produce the box plots.
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123
NUMERICAL DESCRIPTIVE TECHNIQUES
EXCEL
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
17
53
54
55
56
57
58
59
60
61
62
63
64
65
A
B
C
D
E
Popeye’s
Smallest = 112
Q1 = 156.75
Median = 175
Q3 = 192.75
Largest = 238
IQR = 36
Outliers:
F
G
H
295
345
295
345
295
345
295
345
295
345
I
Box Plot
95
145
195
Wendy’s
Smallest = 95
Q1 = 133
Median = 143.5
Q3 = 155
Largest = 201
IQR = 22
Outliers: 201, 199, 190, 97, 95,
245
Box Plot
95
145
195
McDonald’s
Smallest = 121
Q1 = 136
Median = 153
Q3 = 177.5
Largest = 223
IQR = 41.5
Outliers:
245
Box Plot
95
145
195
Hardee’s
Smallest = 121
Q1 = 141.25
Median = 163
Q3 = 207.25
Largest = 338
IQR = 66
Outliers: 338,
245
Box Plot
95
145
195
Jack in Box
Smallest = 190
Q1 = 253.25
Median = 276.5
Q3 = 297.5
Largest = 355
IQR = 44.25
Outliers:
245
Box Plot
95
145
195
245
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MINITAB
Box Plot of Popeye’s, Wendy’s, McDonald’s, Hardee’s, Jack in Box
Popeye’s
Wendy’s
McDonald’s
Hardee’s
Jack in Box
100
150
200
250
300
350
Data
INTERPRET
Wendy’s times appear to be the lowest and most consistent. The service times for
Hardee’s display considerably more variability. The slowest service times are provided
by Jack in the Box. The service times for Popeye’s, Wendy’s, and Jack in the Box seem
to be symmetric. However, the times for McDonald’s and Hardee’s are positively
skewed.
Measures of Relative Standing and Variability for Ordinal Data
Because the measures of relative standing are computed by ordering the data, these statistics are appropriate for ordinal as well as for interval data. Furthermore, because the
interquartile range is calculated by taking the difference between the upper and lower
quartiles, it too can be employed to measure the variability of ordinal data.
Here are the factors that tell us when to use the techniques presented in this
section.
Factors That Identify When to Compute Percentiles and Quartiles
1. Objective: Describe a single set of data
2. Type of data: Interval or ordinal
3. Descriptive measurement: Relative standing
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125
Factors That Identify When to Compute the Interquartile Range
1. Objective: Describe a single set of data
2. Type of data: Interval or ordinal
3. Descriptive measurement: Variability
EXERCISES
4.40 Calculate the first, second, and third quartiles of the
following sample.
5 8 2 9 5 3 7 4 2 7 4 10 4 3 5
4.41 Find the third and eighth deciles (30th and 80th per-
centiles) of the following data set.
26
22
23
15
29
31
31
30
4.50 Xr04-50 Accountemps, a company that supplies tem-
24
20
4.42 Find the first and second quintiles (20th and 40th
percentiles) of the data shown here.
52
71
61
39
88
73
43
51
64
60
4.43 Determine the first, second, and third quartiles of
the following data.
10.5
14.1
14.7
13.9
15.3
18.5
17.7
13.9
15.9
15.1
12.2
14.7
10.0
4.44 Calculate the 3rd and 6th deciles of the accompany-
ing data.
7
4
porary workers, sponsored a survey of 100 executives.
Each was asked to report the number of minutes they
spend screening each job resume they receive.
a. Compute the quartiles.
b. What information did you derive from the quartiles? What does this suggest about writing your
resume?
4.51 Xr04-51 How much do pets cost? A random sample of
dog and cat owners was asked to compute the
amounts of money spent on their pets (exclusive of
pet food). Draw a box plot for each data set and
describe your findings.
4.52 Xr04-52 The Travel Industry Association of America
18 12
10 21
17
5
29
8
18
4
27 30 2
4.45 Refer to Exercise 4.43. Determine the interquartile
range.
4.46 Refer to Exercise 4.40. Determine the interquartile
range.
4.47 Compute the interquartile range from the following
data.
5
expert has stated that he thinks that most drivers
drive at speeds that they consider safe. He suggested
that the “correct” speed limit should be set at the
85th percentile. Suppose that a random sample of
400 speeds on a highway where the limit is 60 mph
was recorded. Find the “correct” speed limit.
8
14
6
21
11
9
10
18
2
4.48 Draw the box plot of the following set of data.
9 28 15
20 23 31
21 12
11 19
22
24
29
16 13
The following exercises require a computer and software.
4.49 Xr04-49 Many automotive experts believe that speed
limits on highways are too low. One particular
sponsored a poll that asked a random sample of people how much they spent in preparation for pleasure
travel. Determine the quartiles and describe what
they tell you.
4.53 Xr04-53 The career-counseling center at a university
wanted to learn more about the starting salaries of
the university’s graduates. They asked each graduate
to report the highest salary offer received. The survey also asked each graduate to report the degree
and starting salary (column 1 = BA, column 2 = BSc,
column 3 = BBA, column 4 = other). Draw box plots
to compare the four groups of starting salaries.
Report your findings.
4.54 Xr04-54 A random sample of Boston Marathon run-
ners was drawn and the times to complete the race
were recorded.
a. Draw the box plot.
b. What are the quartiles?
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CHAPTER 4
a. Calculate the quartiles of these data.
b. What do these statistics tell you about the
amount of time spent in this restaurant?
c. Identify outliers.
d. What information does the box plot deliver?
4.55 Xr04-55 Do golfers who are members of private
courses play faster than players on a public course?
The amount of time taken for a sample of privatecourse and public-course golfers was recorded.
a. Draw box plots for each sample.
b. What do the box plots tell you?
4.57 Xr04-57 In the United States, taxpayers are
4.56 Xr04-56 For many restaurants, the amount of time cus-
tomers linger over coffee and dessert negatively affect
profits. To learn more about this variable, a sample of
200 restaurant groups was observed, and the amount of
time customers spent in the restaurant was recorded.
allowed to deduct mortgage interest from their
incomes before calculating the amount of income
tax they are required to pay. In 2005, the Internal
Revenue Service sampled 500 tax returns that had
a mortgage-interest deduction. Calculate the
quartiles and describe what they tell you.
(Adapted from Statistical Abstract of the United
States, 2009, Table 471.)
American National Election Survey Exercises
4.58 ANES2008* In the 2008 survey, people were
asked to indicate the amount of time they spent
in a typical day receiving news about the election
on the Internet (TIME1) and on television
(TIME2). Compare the two amounts of time by
drawing box plots (using the same scale) and
describe what the graphs tell you. (Excel users:
You must have adjacent columns.
We recommend that you copy the two columns
into adjacent columns in a separate spreadsheet.)
4.59 ANES2008* Draw a box plot of the ages
(AGE) of respondents in the 2008 survey.
4.60 ANES2008* Draw a box plot of the
education level of both married spouses (EDUC
and SPEDUC). Describe your findings.
General Social Survey Exercises
4.61 GSS2008* Draw a box plot of the ages
(AGE) of respondents from the 2008 survey.
Briefly describe the graph.
4 . 4 MEASURES
OF
4.62 GSS2008* Produce a box plot of the
amount of television watched (TVHOURS). State
what the graph tells you.
L I N E A R R E L AT I O N S H I P
In Chapter 3, we introduced the scatter diagram, a graphical technique that describes
the relationship between two interval variables. At that time, we pointed out that we
were particularly interested in the direction and strength of the linear relationship. We
now present three numerical measures of linear relationship that provide this information: covariance, coefficient of correlation, and coefficient of determination. Later in this section we discuss another related numerical technique, the least squares line.
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127
Covariance
As we did in Chapter 3, we label one variable X and the other Y.
Covariance
N
a (xi - mx)(yi - my)
Population covariance: sxy =
i=1
N
n
a (xi - x)(yi - y)
Sample covariance: sxy =
i=1
n - 1
The denominator in the calculation of the sample covariance is n ⫺ 1, not the
more logical n for the same reason we divide by n ⫺ 1 to calculate the sample variance
(see page 109). If you plan to compute the sample covariance manually, here is a shortcut calculation.
Shortcut for Sample Covariance
sxy =
n
1
D a xiyi n - 1 i=1
n
n
i=1
i=1
a xi a yi
n
T
To illustrate how covariance measures the linear relationship, examine the following three sets of data.
Set 1
xi
yi
(xi ⴚⴚ
x)
(yi ⴚⴚ
y)
(xi ⴚⴚ
x )(yi ⴚⴚ
y)
2
13
–3
–7
21
6
20
1
0
0
7
27
2
7
14
x = 5
y = 20
sxy = 35/2 = 17.5
Set 2
xi
2
yi
27
(xi ⴚⴚ
x)
–3
(yi ⴚⴚ
y)
7
(xi ⴚⴚ
x )(yi ⴚⴚ
y)
–21
6
20
1
0
0
7
13
2
–7
–14
x = 5
y = 20
sxy = - 35/2 = - 17.5
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CHAPTER 4
Set 3
xi
yi
(xi ⴚⴚ
x)
(yi ⴚⴚ
y)
(xi ⴚⴚ
x )(yi ⴚⴚ
y)
2
20
–3
0
0
6
27
1
7
7
13
2
–7
x = 5
y = 20
7
–14
sxy = - 7/2 = - 3.5
Notice that the values of x are the same in all three sets and that the values of y are
also the same. The only difference is the order of the values of y.
In set 1, as x increases so does y. When x is larger than its mean, y is at least as large
as its mean. Thus (xi - x) and (yi - y) have the same sign or 0. Their product is also
positive or 0. Consequently, the covariance is a positive number. Generally, when two
variables move in the same direction (both increase or both decrease), the covariance
will be a large positive number.
If you examine set 2, you will discover that as x increases, y decreases. When x is
larger than its mean, y is less than or equal to its mean. As a result when (xi - x) is positive, (yi - y) is negative or 0. Their products are either negative or 0. It follows that the
covariance is a negative number. In general, when two variables move in opposite directions, the covariance is a large negative number.
In set 3, as x increases, y does not exhibit any particular direction. One of the products
(xi - x) (yi - y) is 0, one is positive, and one is negative. The resulting covariance is a small
number. In general, when there is no particular pattern, the covariance is a small number.
We would like to extract two pieces of information. The first is the sign of the
covariance, which tells us the nature of the relationship. The second is the magnitude,
which describes the strength of the association. Unfortunately, the magnitude may be
difficult to judge. For example, if you’re told that the covariance between two variables
is 500, does this mean that there is a strong linear relationship? The answer is that it is
impossible to judge without additional statistics. Fortunately, we can improve on the
information provided by this statistic by creating another one.
Coefficient of Correlation
The coefficient of correlation is defined as the covariance divided by the standard
deviations of the variables.
Coefficient of Correlation
Population coefficient of correlation: r =
Sample coefficient of correlation: r =
sxy
sxsy
sxy
sxsy
The population parameter is denoted by the Greek letter rho.
The advantage that the coefficient of correlation has over the covariance is that the
former has a set lower and upper limit. The limits are ⫺1 and ⫹1, respectively—that is,
-1 … r … +1
and
-1 … r … + 1
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NUMERICAL DESCRIPTIVE TECHNIQUES
129
When the coefficient of correlation equals –1, there is a negative linear relationship
and the scatter diagram exhibits a straight line. When the coefficient of correlation
equals ⫹1, there is a perfect positive relationship. When the coefficient of correlation
equals 0, there is no linear relationship. All other values of correlation are judged in
relation to these three values. The drawback to the coefficient of correlation is that—
except for the three values ⫺1, 0, and ⫹1—we cannot interpret the correlation. For
example, suppose that we calculated the coefficient of correlation to be ⫺.4. What
does this tell us? It tells us two things. The minus sign tells us the relationship is negative and because .4 is closer to 0 than to 1, we judge that the linear relationship is
weak. In many applications, we need a better interpretation than the “linear relationship is weak.” Fortunately, there is yet another measure of the strength of a linear relationship, which gives us more information. It is the coefficient of determination, which
we introduce later in this section.
EXAMPLE
4.16
Calculating the Coefficient of Correlation
Calculate the coefficient of correlation for the three sets of data on pages 126–127.
SOLUTION
Because we’ve already calculated the covariances we need to compute only the standard
deviations of X and Y.
x =
2 + 6 + 7
= 5.0
3
13 + 20 + 27
= 20.0
3
(2 - 5)2 + (6 - 5)2 + (7 - 5)2
9 + 1 + 4
=
= 7.0
s2x =
3 - 1
2
(13 - 20)2 + (20 - 20)2 + (27 - 20)2
49 + 0 + 49
s2y =
=
= 49.0
3 - 1
2
y =
The standard deviations are
sx = 27.0 = 2.65
sy = 249.0 = 7.00
The coefficients of correlation are:
sxy
17.5
= .943
=
Set 1: r =
sxsy
(2.65)(7.0)
sxy
-17.5
= - .943
=
Set 2: r =
sxsy
(2.65)(7.0)
sxy
-3.5
= - .189
=
Set 3: r =
sxsy
(2.65)(7.0)
It is now easier to see the strength of the linear relationship between X and Y.
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Comparing the Scatter Diagram, Covariance, and Coefficient of
Correlation
The scatter diagram depicts relationships graphically; the covariance and the coefficient of
correlation describe the linear relationship numerically. Figures 4.2, 4.3, and 4.4 depict
three scatter diagrams. To show how the graphical and numerical techniques compare, we
calculated the covariance and the coefficient of correlation for each. (The data are stored in
files Fig04-02, Fig04-03, and Fig04-04.) As you can see, Figure 4.2 depicts a strong positive relationship between the two variables. The covariance is 36.87, and the coefficient of
correlation is .9641. The variables in Figure 4.3 produced a relatively strong negative linear relationship; the covariance and coefficient of correlation are –34.18 and –.8791,
respectively. The covariance and coefficient of correlation for the data in Figure 4.4 are
2.07 and .1206, respectively. There is no apparent linear relationship in this figure.
4.2
FIGURE
Strong Positive Linear Relationship
Y
25
20
15
10
5
X
0
5
10
15
20
25
–5
4.3
FIGURE
Strong Negative Linear Relationship
Y
15
10
5
X
0
5
–5
10
15
20
25
–10
–15
–20
FIGURE
4.4
No Linear Relationship
Y
12
10
8
6
4
2
X
0
–2
5
10
15
20
25
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131
S E E I N G S TAT I S T I C S
applet 1 Scatter Diagrams and Correlation
In Section 1.3, we introduced applets as
a method to show students of applied
statistics how statistical techniques
work and gain insights into the underlying principles. The applets are stored on
Keller’s website that accompanies this
book. See the README file for instructions on how to use them.
Instructions for Applet 1
Use your mouse to move the slider in
the graph. As you move the slider,
observe how the coefficient of correlation changes as the points become more
“organized” in the scatter diagram. If
you click Switch sign, you can see the
difference between positive and negative coefficients. The following figures
displays the applet for two values of r.
Describe the pattern of the data
points.
1.2 Drag the slider to the left until the
a pattern? Or do the points appear
to be scattered randomly?
1.4 Drag the slider until the correlation
correlation coefficient r is ⫺1.0.
coefficient r is .5 (approximately).
Describe the pattern of the data
Can you detect a pattern? Now click
points. In what way does it differ
on the Switch Sign button to
from the case where r ⫽ 1.0?
change the correlation coefficient r
1.3 Drag the slider toward the center
to ⫺.5. How does the pattern
Applet Exercises
until the correlation coefficient
change when the sign switches?
1.1 Drag the slider to the right until the
r is 0 (approximately). Describe the
Switch back and forth several times
pattern of the data points. Is there
so you can see the changes.
correlation coefficient r is 1.0.
S E E I N G S TAT I S T I C S
applet 2 Scatter Patterns and Correlation
This applet allows you to place points
clear all points. The figure shown here
on a graph and see the resulting value
depicts a scatter diagram and its coeffi-
of the coefficient of correlation.
cient of correlation.
Instructions
Applet Exercises
Click on the graph to place a point. As
2.1 Create a scatter diagram where r is
you add points, the correlation coeffi-
approximately 0. Describe how you
cient is recalculated. Click to add points
did it.
in various patterns to see how the cor-
2.2 Create a scatter diagram where r is
relation does (or does not) reflect those
approximately 1. Describe how this
patterns. Click on the Reset button to
was done.
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2.3 Plot points such that r is
2.4 Plot the points on a scatter diagram
2.5 Repeat Applet Exercise 2.4, adding
approximately .5. How would you
where r is approximately 1. Now
two points. How close to r = 0 did
describe the resulting scatter
add one more point, decreasing r by
you get?
diagram?
as much as possible. What does this
tell you about extreme points?
Least Squares Method
When we presented the scatter diagram in Section 3.3, we pointed out that we were
interested in measuring the strength and direction of the linear relationship. Both can
be more easily judged by drawing a straight line through the data. However, if different
people draw a line through the same data set, it is likely that each person’s line will differ from all the others. Moreover, we often need to know the equation of the line.
Consequently, we need an objective method of producing a straight line. Such a method
has been developed; it is called the least squares method.
The least squares method produces a straight line drawn through the points so that
the sum of squared deviations between the points and the line is minimized. The line is
represented by the equation:
yN = b0 + b1x
where b0 is the y-intercept (where the line intercepts the y-axis), and b1 is the slope
(defined as rise/run), and yN (y hat) is the value of y determined by the line. The coefficients
b0 and b1 are derived using calculus so that we minimize the sum of squared deviations:
n
2
a (yi - yN i)
i=1
Least Squares Line Coefficients
b1 =
sxy
s2x
b0 = y - b1x
APPLICATIONS in ACCOUNTING
Breakeven Analysis
Breakeven analysis is an extremely important business tool, one that you will likely encounter
repeatedly in your course of studies. It can used to determine how much sales volume your
business needs to start making a profit.
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133
In Section 3.1 (page 44) we briefly introduced the four P’s of marketing and illustrated the
problem of pricing with Example 3.1. Breakeven analysis is especially useful when managers are
attempting to determine the appropriate price for the company’s products and services.
A company’s profit can be calculated simply as
Profit = (Price per unit – variable cost per unit) * (Number of units sold) – Fixed costs
The breakeven point is the number of units sold such that the profit is 0. Thus, the breakeven
point is calculated as
Number of units sold = Fixed cost / (Price – Variable cost)
Managers can use the formula to help determine the price that will produce a profit. However, to
do so requires knowledge of the fixed and variable costs. For example, suppose that a bakery sells
only loaves of bread. The bread sells for $1.20, the variable cost is $0.40, and the fixed annual
costs are $10,000. The breakeven point is
Number of units sold = 10,000/ (1.20 – 0.40) = 12,500
The bakery must sell more than 12,500 loaves per year to make a profit.
In the next application box, we discuss fixed and variable costs.
APPLICATIONS in ACCOUNTING
Fixed and Variable Costs
© Steve Allen/Brand X Pictures/
Jupiter images
Fixed costs are costs that must be paid whether or not any units are produced.
These costs are “fixed” over a specified period of time or range of production.
Variable costs are costs that vary directly with the number of products produced. For the previous bakery example, the fixed costs would include rent and
maintenance of the shop, wages paid to employees, advertising costs, telephone,
and any other costs that are not related to the number of loaves baked. The variable cost is primarily the cost of ingredients, which rises in relation to the number of
loaves baked.
Some expenses are mixed. For the bakery example, one such cost is the cost of electricity.
Electricity is needed for lights, which is considered a fixed cost, but also for the ovens and other
equipment, which are variable costs.
There are several ways to break the mixed costs into fixed and variable components. One
such method is the least squares line; that is, we express the total costs of some component as
y ⫽ b0 ⫹ b1x
where y ⫽ total mixed cost, b0 ⫽ fixed cost, b1 ⫽ variable cost, and x is the number of units.
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EXAMPLE
4.17
DATA
Xm04-17
Estimating Fixed and Variable Costs
A tool and die maker operates out of a small shop making specialized tools. He is considering increasing the size of his business and needs to know more about his costs. One
such cost is electricity, which he needs to operate his machines and lights. (Some jobs
require that he turn on extra bright lights to illuminate his work.) He keeps track of his
daily electricity costs and the number of tools that he made that day. These data are
listed next. Determine the fixed and variable electricity costs.
Day
Number of tools
Electricity cost
1
2
3
4
5
6
7
7
3
2
5
8
11
5
23.80 11.89 15.98 26.11 31.79 39.93 12.27
8
15
40.06
9
3
21.38
10
6
18.65
SOLUTION
The dependent variable is the daily cost of electricity, and the independent variable is
the number of tools. To calculate the coefficients of the least squares line and other statistics (calculated below), we need the sum of X, Y, XY, X2, and Y2.
Day
X
1
2
3
4
5
6
7
8
9
10
7
3
2
5
8
11
5
15
3
6
23.80
11.89
15.98
26.11
31.79
39.93
12.27
40.06
21.38
18.65
Total
65
241.86
X2
Y2
166.60
35.67
31.96
130.55
254.32
439.23
61.35
600.90
64.14
111.90
49
9
4
25
64
121
25
225
9
36
566.44
141.37
255.36
681.73
1010.60
1594.40
150.55
1604.80
457.10
347.82
1896.62
567
6810.20
Y
XY
Covariance:
sxy =
n
1
D a xi yi n - 1 i=1
n
n
i=1
i=1
a xi a yi
n
T =
(65)(241.86)
1
c1896.62 d = 36.06
10 - 1
10
Variance of X:
n
s2x =
n
1
E a x2i n - 1 i=1
¢ a xi ≤
i=1
n
2
U =
(65)2
1
c567 d = 16.06
10 - 1
10
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135
Sample means
x =
65
a xI
=
= 6.5
n
10
y =
241.86
a yi
=
= 24.19
n
10
The coefficients of the least squares line are
Slope
b1 =
sxy
s2x
=
36.06
= 2.25
16.06
y-intercept:
b0 = y - b1x = 24.19 - (2.25)(6.5) = 9.57
The least squares line is
yN = 9.57 + 2.25x
EXCEL
Electrical costs
50
y = 2.2459x + 9.5878
40
30
20
10
0
0
2
4
6
8
10
12
14
16
Number of tools
INSTRUCTIONS
1. Type or import the data into two columns where the first column stores the values of
X and the second stores Y. (Open Xm04-17.) Highlight the columns containing the
variables. Follow the instructions to draw a scatter diagram (page 75).
2. In the Chart Tools and Layout menu, click Trendline and Linear Trendline.
3. Click Trendline and More Trendline Options . . . . Click Display Equation on
Chart.
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CHAPTER 4
MINITAB
Fitted Line Plot
Electrical costs = 9.588 + 2.246 Number of tools
5.38185
S
75.9%
R-Sq
R-Sq(adj) 72.9%
Electrical costs
45
40
35
30
25
20
15
10
2
4
6
8
10
12
14
16
Number of tools
INSTRUCTIONS
1. Type or import the data into two columns. (Open Xm04-17.)
2. Click Stat, Regression, and Fitted Line Plot.
3. Specify the Response [Y] (Electrical cost) and the Predictor [X] (Number of tools)
variables. Specify Linear.
INTERPRET
The slope is defined as rise/run, which means that it is the change in y (rise) for a oneunit increase in x (run). Put less mathematically, the slope measures the marginal rate of
change in the dependent variable. The marginal rate of change refers to the effect of
increasing the independent variable by one additional unit. In this example, the slope is
2.25, which means that in this sample, for each one-unit increase in the number of
tools, the marginal increase in the electricity cost is $2.25. Thus, the estimated variable
cost is $2.25 per tool.
The y-intercept is 9.57; that is, the line strikes the y-axis at 9.57. This is simply the
value of yN when x ⫽ 0. However, when x ⫽ 0, we are producing no tools and hence the
estimated fixed cost of electricity is $9.57 per day.
Because the costs are estimates based on a straight line, we often need to know how
well the line fits the data.
EXAMPLE
4.18
DATA
Xm04-17
Measuring the Strength of the Linear Relationship
Calculate the coefficient of correlation for Example 4.17.
SOLUTION
To calculate the coefficient of correlation, we need the covariance and the standard
deviations of both variables. The covariance and the variance of X were calculated in
Example 4.17. The covariance is
sxy ⫽ 36.06
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137
and the variance of X is
sx2 ⫽ 16.06
Standard deviation of X is
sx = 2s2x = 216.06 = 4.01
All we need is the standard deviation of Y.
n
s2y =
n
1
E a y2i n - 1 i=1
¢ a yi ≤
2
i=1
n
U =
(241.86)2
1
c6810.20 d = 106.73
10 - 1
10
sy = 2s2y = 2106.73 = 10.33
The coefficient of correlation is
r =
sXY
sxsy
=
36.06
= .8705
(4.01)(10.33)
EXCEL
As with the other statistics introduced in this chapter, there is more than one way to calculate the coefficient of correlation and the covariance. Here are the instructions for
both.
INSTRUCTIONS
1. Type or import the data into two columns. (Open Xm04-17.) Type the following into
any empty cell.
⫽ CORREL([Input range of one variable], [Input range of second variable])
In this example, we would enter
= CORREL(B1:B11, C1:C11)
To calculate the covariance, replace CORREL with COVAR.
Another method, which is also useful if you have more than two variables and would
like to compute the coefficient of correlation or the covariance for each pair of variables,
is to produce the correlation matrix and the variance–covariance matrix. We do the correlation matrix first.
A
1
2
3
Number of tools
Electrical costs
B
Number of tools
1
0.8711
C
Electrical costs
1
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INSTRUCTIONS
1. Type or import the data into adjacent columns. (Open Xm04-17.)
2. Click Data, Data Analysis, and Correlation.
3. Specify the Input Range (B1:C11).
The coefficient of correlation between number of tools and electrical costs is .8711
(slightly different from the manually calculated value). (The two 1s on the diagonal of the
matrix are the coefficients of number of tools and number of tools, and electrical costs
and electrical costs, telling you the obvious.)
Incidentally, the formula for the population parameter r (Greek letter rho) and for
the sample statistic r produce exactly the same value.
The variance–covariance matrix is shown next.
A
1
2
3
B
Number of tools
14.45
32.45
Number of tools
Electrical costs
C
Electrical costs
96.06
INSTRUCTIONS
1. Type or import the data into adjacent columns. (Open Xm04-17.)
2. Click Data, Data Analysis, and Covariance.
3. Specify the Input Range (B1:C11).
Unfortunately, Excel computes the population parameters. In other words,
the variance of the number of tools is s2x = 14.45, the variance of the electrical costs
is s2y = 96.06, and the covariance is sxy = 32.45. You can convert these parameters to
statistics by multiplying each by n/(n – 1).
D
1
2
3
Number of tools
Electrical costs
E
Number of tools
16.06
36.06
F
Electrical costs
106.73
MINITAB
Correlations: Number of tools, Electrical costs
Pearson correlation of Number of tools and Electrical costs = 0.871
INSTRUCTIONS
1. Type or import the data into two columns. (Open Xm04-17.)
2. Click Calc, Basic Statistics and Correlation . . . .
3. In the Variables box, type Select the variables (Number of Tools, Electrical
Costs).
Covariances: Number of tools, Electrical costs
Number of tools
Electrical costs
Number of tools
16.0556
36.0589
Electrical costs
106.7301
INSTRUCTIONS
Click Covariance . . . instead of Correlation . . . in step 2 above.
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139
INTERPRET
The coefficient of correlation is .8711, which tells us that there is a positive linear relationship between the number of tools and the electricity cost. The coefficient of correlation tells us that the linear relationship is quite strong and thus the estimates of the
fixed and variable costs should be good.
Coefficient of Determination
When we introduced the coefficient of correlation (page 128), we pointed out that
except for ⫺1, 0, and ⫹1 we cannot precisely interpret its meaning. We can judge the
coefficient of correlation in relation to its proximity to only ⫺1, 0, and ⫹1. Fortunately,
we have another measure that can be precisely interpreted. It is the coefficient of determination, which is calculated by squaring the coefficient of correlation. For this reason,
we denote it R2 .
The coefficient of determination measures the amount of variation in the dependent variable that is explained by the variation in the independent variable. For example, if the coefficient of correlation is ⫺1 or ⫹1, a scatter diagram would display all the
points lining up in a straight line. The coefficient of determination is 1, which we interpret to mean that 100% of the variation in the dependent variable Y is explained by the
variation in the independent variable X. If the coefficient of correlation is 0, then there
is no linear relationship between the two variables, R2 ⫽ 0, and none of the variation in
Y is explained by the variation in X. In Example 4.18, the coefficient of correlation was
calculated to be r ⫽ .8711. Thus, the coefficient of determination is
r2 ⫽ (.8711)2 ⫽ .7588
This tells us that 75.88% of the variation in electrical costs is explained by the number
of tools. The remaining 24.12% is unexplained.
Using the Computer
EXCEL
You can use Excel to calculate the coefficient of correlation and then square the result.
Alternatively, use Excel to draw the least squares line. After doing so, click Trendline,
Trendline Options, and Display R-squared value on chart.
MINITAB
Minitab automatically prints the coefficient of determination.
The concept of explained variation is an extremely important one in statistics. We
return to this idea repeatedly in Chapters 13, 14, 16, 17, and 18. In Chapter 16, we
explain why we interpret the coefficient of determination in the way that we do.
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To determine the cost of an additional win, we must describe the relationship between two variables. To do so, we use the least squares method to produce a straight line through the data.
Because we believe that the number of games a baseball team wins depends to some extent on its
team payroll, we label Wins as the dependent variable and Payroll as the independent variable.
Because of rounding problems, we expressed the payroll in the number of millions of dollars.
© AP Photo/Charles Krupa
Cost of One More Win: Solution
EXCEL
120
100
Wins
80
y = 0.1725x + 65.758
R2 = 0.2512
60
40
20
0
0.000
50.000
100.000
150.000
200.000
250.000
Payroll
As you can see, Excel outputs the least squares line and the coefficient of determination.
MINITAB
Fitted Line Plot
Wins = 65.76 + 0.1725 Payroll ($millions)
100
Wins
90
80
70
S
R-Sq
R-Sq(adj)
60
50
75
100
125
150
175
10.0703
25.1%
22.4%
200
Payroll ($millions)
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141
INTERPRET
The least squares line is
yN = 65.758 + .1725 x
The slope is equal to .1725, which is the marginal rate of change in games won for
each one-unit increase in payroll. Because payroll is measured in millions of dollars,
we estimate that for each $1 million increase in the payroll, the number of games
won increases on average by .1725. Thus, to win one more game requires on average
an additional expenditure of an incredible $5,797,101 (calculated as 1 million/.1725).
Besides analyzing the least squares line, we should determine the strength of the
linear relationship. The coefficient of determination is .2512, which means that the
variation in the team’s payroll explains 25.12% of the variation in the team’s number
of games won. This suggests that some teams win a small number of games with
large payrolls, whereas others win a large number of games with small payrolls. In
the next section, we will return to this issue and examine why some teams perform
better than predicted by the least squares line.
Interpreting Correlation
Because of its importance, we remind you about the correct interpretation of the analysis of the relationship between two interval variables that we discussed in Chapter 3. In
other words, if two variables are linearly related, it does not mean that X causes Y. It
may mean that another variable causes both X and Y or that Y causes X. Remember
Correlation is not Causation
We complete this section with a review of when to use the techniques introduced in
this section.
Factors That Identify When to Compute Covariance, Coefficient of
Correlation, Coefficient of Determination, and Least Squares Line
1. Objective: Describe the relationship between two variables
2. Type of data: Interval
EXERCISES
4.63 The covariance of two variables has been calculated
to be –150. What does the statistic tell you about the
relationship between the two variables?
b. Calculate the coefficient of determination and
describe what this says about the relationship
between the two variables.
4.64 Refer to Exercise 4.63. You’ve now learned that the
4.65 Xr04-65 A retailer wanted to estimate the monthly
two sample standard deviations are 16 and 12.
a. Calculate the coefficient of correlation. What
does this statistic tell you about the relationship
between the two variables?
fixed and variable selling expenses. As a first step,
she collected data from the past 8 months. The total
selling expenses ($1,000) and the total sales ($1,000)
were recorded and listed below.
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Total Sales
Selling Expenses
20
40
60
50
50
55
60
70
14
16
18
17
18
18
18
20
The following exercises require a computer and software.
4.68 Xr04-68 The unemployment rate is an important
a. Compute the covariance, the coefficient of correlation, and the coefficient of determination and
describe what these statistics tell you.
b. Determine the least squares line and use it to
produce the estimates the retailer wants.
4.66 Xr04-66 Are the marks one receives in a course
related to the amount of time spent studying the
subject? To analyze this mysterious possibility, a
student took a random sample of 10 students who
had enrolled in an accounting class last semester.
He asked each to report his or her mark in the
course and the total number of hours spent studying
accounting. These data are listed here.
Marks
77 63 79 86 51 78 83 90 65 47
Time spent
studying 40 42 37 47 25 44 41 48 35 28
a.
b.
c.
d.
e.
Calculate the covariance.
Calculate the coefficient of correlation.
Calculate the coefficient of determination.
Determine the least squares line.
What do the statistics calculated above tell you
about the relationship between marks and study
time?
4.67 Xr04-67 Students who apply to MBA programs must
take the Graduate Management Admission Test
(GMAT). University admissions committees use the
GMAT score as one of the critical indicators of how
well a student is likely to perform in the MBA program. However, the GMAT may not be a very strong
indicator for all MBA programs. Suppose that an
MBA program designed for middle managers who
wish to upgrade their skills was launched 3 years ago.
To judge how well the GMAT score predicts MBA
performance, a sample of 12 graduates was taken.
Their grade point averages in the MBA program
(values from 0 to 12) and their GMAT score (values
range from 200 to 800) are listed here. Compute the
covariance, the coefficient of correlation, and the
coefficient of determination. Interpret your findings.
GMAT and GPA Scores for 12 MBA Students
GMAT
MBA GPA
599
9.6
689
8.8
584
7.4
631 594
10.0 7.8
643
9.2
GMAT
MBA GPA
656
9.6
594
8.4
710
11.2
611
7.6
683
8.0
593
8.8
measure of a country’s economic health. The unemployment rate measures the percentage of people
who are looking for work and who are without jobs.
Another way of measuring this economic variable is
to calculate the employment rate, which is the percentage of adults who are employed. Here are the
unemployment rates and employment rates of
19 countries. Calculate the coefficient of determination and describe what you have learned.
Country
Unemployment
Rate
Employment
Rate
6.7
3.6
6.6
7.2
4.3
9.1
8.6
7.9
5.8
3.8
5.0
2.4
5.3
18.2
4.1
13.0
5.1
5.0
4.8
70.7
74.8
59.9
72.0
77.0
68.1
63.2
69.0
55.4
67.3
74.3
65.4
62.3
53.5
72.2
57.5
73.0
72.2
73.1
Australia
Austria
Belgium
Canada
Denmark
Finland
France
Germany
Hungary
Ireland
Japan
Netherlands
New Zealand
Poland
Portugal
Spain
Sweden
United Kingdom
United States
(Source: National Post Business.)
4.69 Xr04-69 All Canadians have government-funded
health insurance, which pays for any medical care
they require. However, when traveling out of the
country, Canadians usually acquire supplementary
health insurance to cover the difference between the
costs incurred for emergency treatment and what
the government program pays. In the United States,
this cost differential can be prohibitive. Until
recently, private insurance companies (such as
BlueCross BlueShield) charged everyone the same
weekly rate, regardless of age. However, because of
rising costs and the realization that older people frequently incur greater medical emergency expenses,
insurers had to change their premium plans. They
decided to offer rates that depend on the age of the
customer. To help determine the new rates, one
insurance company gathered data concerning the
age and mean daily medical expenses of a random
sample of 1,348 Canadians during the previous
12-month period.
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NUMERICAL DESCRIPTIVE TECHNIQUES
a. Calculate the coefficient of determination.
b. What does the statistic calculated in part (a) tell
you?
c. Determine the least squares line.
d. Interpret the coefficients.
e. What rate plan would you suggest?
4.70 Xr04-70 A real estate developer of single-family
dwellings across the country is in the process of
developing plans for the next several years. An analyst for the company believes that interest rates are
likely to increase but remain at low levels. To help
make decisions about the number of homes to
build, the developer acquired the monthly bank
prime rate and the number of new single-family
homes sold monthly (thousands) from 1963 to
2009. (Source: Federal Reserve Statistics and U.S.
Census Bureau.)
Calculate the coefficient of determination. Explain
what this statistic tells you about the relationship
between the prime bank rate and the number of single-family homes sold.
4.71 Xr04-71 When the price of crude oil increases, do oil
companies drill more oil wells? To determine the
strength and nature of the relationship, an economist
recorded the price of a barrel of domestic crude oil
(West Texas crude) and the number of exploratory oil
wells drilled for each month from 1973 to 2009.
Analyze the data and explain what you have discovered. (Source: U.S. Department of Energy.)
4.72
One way of measuring the extent of unemployment is through the help wanted index, which
measures the number of want ads in the nation’s
newspapers. The higher the index, the greater the
demand for workers. Another measure is the unemployment rate among insured workers. An economist
wanted to know whether these two variables are
related and, if so, how. He acquired the help wanted
index and unemployment rates for each month
between 1951 and 2006 (last year available).
Determine the strength and direction of the relationship. (Source: U.S. Department of Labor Statistics.)
Xr04-72
4.73 Xr04-73 A manufacturing firm produces its products
in batches using sophisticated machines and equipment. The general manager wanted to investigate
the relationship between direct labor costs and the
number of units produced per batch. He recorded
the data from the last 30 batches. Determine the
fixed and variable labor costs.
4.74 Xr04-74 A manufacturer has recorded its cost of elec-
tricity and the total number of hours of machine
time for each of 52 weeks. Estimate the fixed and
variable electricity costs.
143
4.75 Xr04-75 The chapter-opening example showed that
there is a linear relationship between a baseball
team’s payroll and the number of wins. This raises
the question, are success on the field and attendance
related? If the answer is no, then profit-driven owners may not be inclined to spend money to improve
their teams. The statistics practitioner recorded the
number of wins and the average home attendance
for the 2009 baseball season.
a. Calculate whichever parameters you wish to help
guide baseball owners.
b. Estimate the marginal number of tickets sold for
each additional game won.
4.76 Xr04-76 Refer to Exercise 4.75. The practitioner also
recorded the average away attendance for each team.
Visiting teams take a share of the gate, so every
owner should be interested in this analysis.
a. Are visiting team attendance figures related to
number of wins?
b. Estimate the marginal number of tickets sold for
each additional game won.
4.77 Xr04-77 The number of wins and payrolls for the
each team in the National Basketball Association
(NBA) in the 2008–2009 season were recorded.
a. Determine the marginal cost of one more win.
b. Calculate the coefficient of determination and
describe what this number tells you.
4.78 Xr04-78 The number of wins and payrolls for each
team in the National Football League (NFL) in the
2009–2010 season were recorded.
a. Determine the marginal cost of one more win.
b. Calculate the coefficient of determination and
describe what this number tells you.
4.79 Xr04-79 The number of wins and payrolls for each
team in the National Hockey League (NHL) in the
2008–2009 season were recorded.
a. Determine the marginal cost of one more win.
b. Calculate the coefficient of determination and
describe what this number tells you.
4.80 Xr04-80 We recorded the home and away attendance
for the NBA for the 2008–2009 season.
a. Analyze the relationship between the number of
wins and home attendance.
b. Perform a similar analysis for away attendance.
4.81 Xr04-81 Refer to Exercise 4.77. The relatively weak
relationship between the number of wins and
home attendance may be explained by the size of
the arena each team plays in. The ratio of home
attendance to the arena’s capacity was calculated.
Is percent of capacity more strongly related to the
number of wins than average home attendance?
Explain.
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4.82 Xr04-82 Analyze the relationship between the num-
4.83 Xr04-83 Repeat Exercise 4.81 for the NFL.
ber of wins and home and away attendance in the
National Football League in the 2009–2010 season.
General Social Survey Exercises
(Excel users: You must have adjacent columns. We
recommend that you copy the two columns into
adjacent columns in a separate spreadsheet.)
b. Measure the strength of the linear relationship
using an appropriate statistic and explain what the
statistic tells you.
4.84 GSS2008* Do more educated people watch
less television?
4.85 GSS2006* Using the 2006 survey, determine
whether the number of years of education
(EDUC) of the respondent is linearly related to
the number of years of education of his or her
father (PAEDUC).
a. To answer this question use the least squares
method to determine how education (EDUC)
affects the amount of time spent watching
television (TVHOURS).
American National Election Survey Exercise
4.86 ANES2008* Determine whether the age
(AGE) of the respondent and the amount of
4 . 5 (O P T I O N A L ) A PPL I C AT I O N S
IN
time he or she watches television news in a
typical week (TIME2) are linearly related.
P RO F E S S I O N A L S P ORT S : B A S E B A L L
In the chapter-opening example, we provided the payrolls and the number of wins from
the 2009 season. We discovered that there is a weak positive linear relationship between
number of wins and payroll. The strength of the linear relationship tells us that some teams
with large payrolls are not successful on the field, whereas some teams with small payrolls
win a large number of games. It would appear that although the amount of money teams
spend is a factor, another factor is how teams spend their money. In this section, we will
analyze the eight seasons between 2002 and 2009 to see how small-payroll teams succeed.
Professional sports in North America is a multibillion-dollar business. The cost of
a new franchise in baseball, football, basketball, and hockey is often in the hundreds of
millions of dollars. Although some teams are financially successful during losing seasons, success on the field is often related to financial success. (Exercises 4.75 and 4.76
reveal that there is a financial incentive to win more games.)
It is obvious that winning teams have better players. But how does a team get better players? Teams acquire new players in three ways:
1.
2.
3.
They can draft players from high school and college.
They can sign free agents on their team or on other teams.
They can trade with other teams.
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145
Drafting Players
Every year, high school and university players are drafted by major league baseball
teams. The order of the draft is in reverse order of the winning percentage the previous
season. Teams that rank low in the standings rank high in the draft. A team that drafts
and signs a player owns the rights to that player for his first 7 years in the minor leagues
and his first 6 years in the major leagues. The decision on whom to draft and in what
order is made by the general manager and a group of scouts who travel the country
watching high school and college games. Young players are often invited to a camp
where variables such as running speed, home run power, and, for pitchers, velocity are
measured. They are often judged by whether a young man “looks” like a player. That is,
taller, more athletic men are judged to be better than shorter, heavier ones.
Free Agency
For the first 3 years in the major leagues, the team can pay a player the minimum, which in
2009 was $400,000 per year. After 3 years, the player is eligible for arbitration. A successful
player can usually increase his salary from $2 million to $3 million through arbitration.
After 6 years, the player can become a free agent and can sign with any major league team.
The top free agents can make well in excess of $10 million per year in a multiyear contract.
Trading
Teams will often trade with each other hoping that the players they acquire will help
them more than the players they traded away. Many trades produce little improvement
in both teams. However, in the history of baseball, there have been some very one-sided
trades that resulted in great improvement in one team and the weakening of the other.
As you can see from the solved chapter-opening example “Cost of One More
Win,” there is a great variation in team payrolls. In 2009, the New York Yankees spent
$201 million, while the Florida Marlins spent $37 million (the amounts listed are payrolls at the beginning of the season). To a very large extent, the ability to finance expensive teams is a function of the city in which the team plays. For example, teams in New
York, Los Angeles, Atlanta, and Arlington, Texas, are in large markets. Tampa Bay,
Oakland, and Minnesota are small-market teams. Large-market teams can afford
higher salaries because of the higher gate receipts and higher fees for local television.
This means that small-market teams cannot compete for the services of the top free
agency players, and thus are more likely to be less successful than large-market teams.
The question arises, can small-market teams be successful on the field and, if so,
how? The answer lies in how players are assessed for the draft and for trades. The decisions about whom to draft, whom to trade for, and whom to give in return are made by
the team’s general manager with the assistance of his assistants and the team’s scouts.
Because scouts are usually former major league and minor league players who were
trained by other former minor league and major league players, they tend to generally
agree on the value of the players in the draft. Similarly, teams making trades often trade
players of “equal” value. As a result, most teams evaluate players in the same way, yielding little differences in a player’s worth. This raises the question, how can a team get the
edge on other teams? The answer lies in statistics.
You won’t be surprised to learn that the two most important variables in determining
the number of wins are the number of runs the team scores and the number of runs the
team allows. The number of runs allowed is a function of the quality of the team’s pitchers and, to a lesser extent, the defense. Most major league teams evaluate pitchers on the
velocity of their fastball. Velocities in the 90 to 100 mile per hour range get the scouts’
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attention. High school and college pitchers with fastball speeds in the 80s are seldom
drafted in the early rounds even when they appear to allow fewer runs by opposing teams.
Scouts also seek out high school and college players with high batting averages and
who hit home runs in high school and college.
The only way that small-budget teams can succeed is for them to evaluate players
more accurately. In practice, this means that they need to judge players differently from
the other teams. In the following analysis, we concentrate on the number of runs a team
scores and the statistics that are related to this variable.
If the scouts are correct in their method of evaluating young players, the variables
that would be most strongly related to the number of runs a team scores are batting average (BA) and the number of home runs (HR). (A player’s batting average is computed by
calculating the number of times the player hits divided by the number of at bats less bases
on balls.) The coefficients of correlation for seasons 2002 to 2009 are listed here.
Year
Coefficients of Correlation
2002 2003
2004
2005 2006 2007 2008
2009
Number of runs & batting average
Number of runs & home runs
.828
.682
.803
.765
.780
.713
.748
.744
.889
.747
.672
.559
.762
.536
.680
.617
Are there better statistics? In other words, are there other team statistics that
correlate more highly with the number of runs a team scores? There are two candidates. The first is the teams’ on-base average (OBA); the second is the slugging percentage (SLG). The OBA is the number of hits plus bases on balls plus being hit by
the pitcher divided by the number of at bats. The SLG is calculated by dividing the
total number of bases (single ⫽ 1, double ⫽ 2, triple ⫽ 3, and home run ⫽ 4) by the
number of at bats minus bases on balls and hit by pitcher. The coefficients of correlation are listed here.
Year
Coefficients of Correlation
2002 2003 2004 2005 2006 2007 2008 2009
Number of runs and on-base average
.835
Number of runs and slugging percentage .913
.916
.951
.875
.929
.783
.790
.800
.854
.875
.885
.834
.903
.851
.911
As you can see, for all eight seasons the OBA had a higher correlation with runs
than did the BA.
Comparing the coefficients of correlation of runs with HR and SLG, we can see
that in all eight seasons SLG was more strongly correlated than was HR.
As we’ve pointed out previously, we cannot definitively conclude a causal relationship from these statistics. However, because most decisions are based on the BA and
HR, these statistics suggest that general managers should place much greater weight on
batters’ ability to get on base instead of simply reading the batting averages.
The Oakland Athletics (and a Statistics) Success Story*
From 2002 to 2006, no team was as successful as the Oakland Athletics in converting a
small payroll into a large number of wins. In 2002, Oakland’s payroll was $40 million and
the team won 103 games. In the same season, the New York Yankees spent $126 million
and won the same number of games. In 2003, Oakland won 96 games, second to
*The Oakland success story is described in the book Moneyball : The Art of Winning an Unfair Game by
Michael Lewis, New York London: W.W. Norton
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New York’s 101 games. Oakland’s payroll was $50 million, whereas New York’s was $153
million. In 2004, Oakland won 91 games with a payroll of $59 million, and the Yankees
won 101 games with a payroll of $184 million. Oakland won 88 games in 2005, and the
Yankees won 95 games. Payrolls were Oakland $55 million, Yankees $208 million. In
2006, the team payrolls were Oakland $62 million, Yankees $199 million. Team wins
were Oakland 93, Yankees 97.
The Athletics owe their success to general managers who were willing to
rethink how teams win. The first of these general managers was Sandy Alderson,
who was hired by Oakland in 1993. He was a complete outsider with no baseball
experience. This was an unusual hire in an organization in which managers and
general managers are either former players or individuals who worked their way up
the organization after years of service in a variety of jobs. At the time, Oakland was
a high-payroll team that was successful on the field. It was in the World Series in
1988, 1989, and 1990, and had the highest payroll in 1991. The team was owned by
Walter A. Haas, Jr., who was willing to spend whatever was necessary to win. When
he died in 1995, the new owners decided that the payroll was too large and limited
it. This forced Alderson to rethink strategy.
Sandy Alderson was a lawyer and a former marine. Because he was an outsider, he
approached the job in a completely different way. He examined each aspect of the game
and, among other things, concluded that before three outs everything was possible, but
after three outs nothing was possible. This led him to the conclusion that the way to
score runs is to minimize each player’s probability of making an out. Rather than judge
a player by his batting average, which is the way every other general manger assessed
players, it would make more sense to judge the player on his on-base average. The onbase average (explained previously) is the probability of not making an out. Thus was
born something quite rare in baseball—a new idea.
Alderson’s replacement is Billy Beane, who continued and extended Alderson’s
thinking, including hiring a Harvard graduate to help manage the statistics.
In the previous edition of this book, we asked the question, why don’t other teams
do the same? The answer in 2010 is that other teams have. In the last three years,
Oakland has not been anywhere nearly as successful as it was in the previous five (winning about 75 games each year). Apparently, Oakland’s approach to evaluating players
has influenced other teams. However, there is still a weak linear relationship between
team success and team payroll. This means that other variables affect how many games
a team wins besides what the team pays its players. Perhaps some clever general manager will find these variables. If he or she does, it may be best not to publicize the discovery in another book.
4 . 6 (O P T I O N A L ) A PPL I C AT I O N S
IN
FINANCE: MARKET MODEL
In the Applications in Finance box in Chapter 3 (page 52), we introduced the terms
return on investment and risk. We described two goals of investing. The first is to maximize the expected or mean return and the second is to minimize the risk. Financial analysts use a variety of statistical techniques to achieve these goals. Most investors are risk
averse, which means that for them minimizing risk is of paramount importance. In
Section 4.2, we pointed out that variance and standard deviation are used to measure
the risk associated with investments.
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APPLICATIONS in FINANCE
Stock Market Indexes
Stock markets such as the New York Stock Exchange (NYSE), NASDAQ, Toronto Stock Exchange (TSE),
and many others around the world calculate indexes to provide information about the prices of stocks
on their exchanges. A stock market index is composed of a number of stocks that more or less represent the entire market. For example, the Dow Jones Industrial Average (DJIA) is the average price of a
group of 30 NYSE stocks of large publicly traded companies. The Standard and Poor’s 500 Index (S&P)
is the average price of 500 NYSE stocks. These indexes represent their stock exchanges and give readers a quick view of how well the exchange is doing as well as the economy of the country as a whole.
The NASDAQ 100 is the average price of the 100 largest nonfinancial companies on the NASDAQ
exchange. The S&P/TSX Composite Index is composed of the largest companies on the TSE.
In this section, we describe one of the most important applications of the use of a least
squares line. It is the well-known and often applied market model. This model assumes that the
rate of return on a stock is linearly related to the rate of return on the stock market index. The
return on the index is calculated in the same way the return on a single stock is computed. For
example, if the index at the end of last year was 10,000 and the value at the end of this year is
11,000, then the market index annual return is 10%. The return on the stock is the dependent
variable Y, and the return on the index is the independent variable X.
We use the least squares line to represent the linear relationship between X and Y. The coefficient b1 is called the stock’s beta coefficient, which measures how sensitive the stock’s rate of return is
to changes in the level of the overall market. For example, if b1 is greater than 1, then the stock’s rate
of return is more sensitive to changes in the level of the overall market than the average stock. To
illustrate, suppose that b1 ⫽ 2. Then a 1% increase in the index results in an average increase of 2% in
the stock’s return. A 1% decrease in the index produces an average 2% decrease in the stock’s return.
Thus, a stock with a beta coefficient greater than 1 will tend to be more volatile than the market.
EXAMPLE
4.19
DATA
Xm04-19
Market Model for Research in Motion
The monthly rates of return for Research in Motion, maker of the BlackBerry (symbol
RIMM), and the NASDAQ index (a measure of the overall NASDAQ stock market)
were recorded for each month between January 2005 and December 2009. Some of
these data are shown below. Estimate the market model and analyze the results.
Month-Year
Index
RIMM
Jan-05
Feb-05
Mar-05
Apr-05
May-05
Jun-05
⫺0.05196
⫺0.00518
⫺0.02558
⫺0.03880
0.07627
⫺0.00544
⫺0.13506
⫺0.07239
0.15563
⫺0.15705
0.28598
⫺0.10902
Jul-09
Aug-09
Sep-09
Oct-09
Nov-09
Dec-09
0.07818
0.01545
0.05642
⫺0.03643
0.04865
0.05808
0.06893
⫺0.03856
⫺0.07432
⫺0.13160
⫺0.01430
0.16670
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SOLUTION
Excel’s scatter diagram and least squares line are shown below. (Minitab produces a similar result.) We added the equation and the coefficient of determination to the scatter
diagram.
0.8
y = 1.9204x + 0.0251
R2 = 0.3865
RIMM Returns
0.6
0.4
0.2
0.0
Index Returns
⫺0.2
⫺0.1
⫺0.2
0.0
0.1
0.2
⫺0.4
⫺0.6
We note that the slope coefficient for RIMM is 1.9204. We interpret this to mean
that for each 1% increase in the NASDAQ index return in this sample, the average
increase in RIMM’s return is 1.9204%. Because b1 is greater than 1, we conclude that
the return on investing in Research in Motion is more volatile and therefore riskier
than the entire NASDAQ market.
Systematic and Firm-Specific Risk
The slope coefficient b1 is a measure of the stock’s market-related (or systematic) risk
because it measures the volatility of the stock price that is related to the overall market
volatility. The slope coefficient only informs us about the nature of the relationship
between the two sets of returns. It tells us nothing about the strength of the linear
relationship.
The coefficient of determination measures the proportion of the total risk that is
market related. In this case, we see that 38.65% of RIMM’s total risk is market
related. That is, 38.65% of the variation in RIMM’s returns is explained by the variation in the NASDAQ index’s returns. The remaining 61.35% is the proportion of the
risk that is associated with events specific to RIMM rather than the market. Financial
analysts (and most everyone else) call this the firm-specific (or nonsystematic) risk. The
firm-specific risk is attributable to variables and events not included in the market
model, such as the effectiveness of RIMM’s sales force and managers. This is the part
of the risk that can be “diversified away” by creating a portfolio of stocks (as will be
discussed in Section 7.3). We cannot, however, diversify away the part of the risk that
is market related.
When a portfolio has been created, we can estimate its beta by averaging the betas
of the stocks that compose the portfolio. If an investor believes that the market is likely
to rise, then a portfolio with a beta coefficient greater than 1 is desirable. Risk-averse
investors or ones who believe that the market will fall will seek out portfolios with betas
less than 1.
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EXERCISES
The following exercises require the use of a computer and software.
4.87
We have recorded the monthly returns for
the S&P 500 index and the following six stocks listed
on the New York Stock Exchange for the period
January 2005 to December 2009.
Xr04-87
AT&T
Aetna
Cigna
Coca-Cola
Disney
Ford
McDonald’s
Bank of Montreal (BMO)
Enbridge
Fortis
Methanex
Research in Motion (RIM)
Telus
Trans Canada Pipeline
Calculate the beta coefficient for each stock and discuss what you have learned about each stock.
4.89 X04-89 We calculated the returns on the NASDAQ
Calculate the beta coefficient for each stock and
briefly describe what it means. (Excel users: To use
the scatter diagram to compute the beta coefficient,
the data must be stored in two adjacent columns.
The first must contain the returns on the index, and
the second stores the returns for whichever stock
whose coefficient you wish to calculate.)
4.88 Xm04-88 Monthly returns for the Toronto Stock
Exchange index and the following stocks on the
Toronto Stock Exchange were recorded for the
years 2005 to 2009.
Barrick Gold
Bell Canada Enterprises (BCE)
4 .7 C O M PA R I N G G R A PH I C A L
index and the following stocks on the NASDAQ
exchange for the period January 2005 to December
2009.
Amazon
Amgen
Apple
Cisco Systems
Google
Intel
Microsoft
Oracle
Research in Motion
Calculate the beta coefficient for each stock and
briefly describe what it means.
AND
NUMERICAL TECHNIQUES
As we mentioned before, graphical techniques are useful in producing a quick picture
of the data. For example, you learn something about the location, spread, and shape of
a set of interval data when you examine its histogram. Numerical techniques provide
the same approximate information. We have measures of central location, measures of
variability, and measures of relative standing that do what the histogram does. The scatter diagram graphically describes the relationship between two interval variables, but so
do the numerical measures covariance, coefficient of correlation, coefficient of determination, and least squares line. Why then do we need to learn both categories of techniques? The answer is that they differ in the information each provides. We illustrate
the difference between graphical and numerical methods by redoing four examples we
used to illustrate graphical techniques in Chapter 3.
EXAMPLE
3. 2
Comparing Returns on Two Investments
In Example 3.2, we wanted to judge which investment appeared to be better. As we
discussed in the Applications in Finance: Return on Investment (page 52), we judge
investments in terms of the return we can expect and its risk. We drew histograms and
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151
attempted to interpret them. The centers of the histograms provided us with information about the expected return and their spreads gauged the risk. However, the histograms were not clear. Fortunately, we can use numerical measures. The mean and
median provide us with information about the return we can expect, and the variance or
standard deviation tell us about the risk associated with each investment.
Here are the descriptive statistics produced by Excel. Minitab’s are similar. (We
combined the output into one worksheet.)
Microsoft Excel Output for Example 3.2
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B
C
Return A
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
D
E
Return B
10.95
3.10
9.88
12.89
21.89
479.35
-0.32
0.54
84.95
-21.95
63
547.27
50
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
12.76
3.97
10.76
#N/A
28.05
786.62
-0.62
0.01
106.47
-38.47
68
638.01
50
We can now see that investment B has a larger mean and median but that investment A has a smaller variance and standard deviation. If an investor were interested in
low-risk investments, then he or she would choose investment A. If you reexamine the
histograms from Example 3.2 (page 53), you will see that the precision provided by the
numerical techniques (mean, median, and standard deviation) provides more useful
information than did the histograms.
3.3
3.4
EXAMPLES
AND
Business Statistics Marks; Mathematical Statistical
Marks
In these examples we wanted to see what differences existed between the marks in the
two statistics classes. Here are the descriptive statistics. (We combined the two printouts in one worksheet.)
Microsoft Excel Output for Examples 3.3 and 3.4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A
B
Marks (Example 3.3)
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(15)
Smallest(15)
C
72.67
1.07
72
67
8.29
68.77
-0.36
0.16
39
53
92
4360
60
79
67
D
E
Marks (Example 3.4)
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(15)
Smallest(15)
66.40
1.610
71.5
75
12.470
155.498
-1.241
-0.217
48
44
92
3984
60
76
53
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The statistics tell us that the mean and median of the marks in the business statistics
course (Example 3.3) are higher than in the mathematical statistics course (Example
3.4). We found that the histogram of the mathematical statistics marks was bimodal,
which we interpreted to mean that this type of approach created differences between
students. The unimodal histogram of the business statistics marks informed us that this
approach eliminated those differences.
Chapter 3 Opening Example
In this example, we wanted to know whether the prices of gasoline and oil were related.
The scatter diagram did reveal a strong positive linear relationship. We can improve on
the quality of this information by computing the coefficient of correlation and drawing
the least squares line.
Excel Output for Chapter 3 Opening Example: Coefficient of Correlation
A
1
2
3
B
Oil
Oil
Gasoline
C
Gasoline
1
0.8574
1
The coefficient of correlation seems to confirm what we learned from the scatter diagram: There is a moderately strong positive linear relationship between the two
variables.
Excel Output for Chapter 3 Opening Example: Least Squares Line
Price of Gasoline
3.500
3.000
y = 0.0293x + 0.6977
R2 = 0.9292
2.500
2.000
1.500
1.000
0.500
0.000
0.00
20.00
40.00
Price of Oil
60.00
80.00
The slope coefficient tells us that for each dollar increase in the price of a barrel of
oil, the price of a (U.S.) gallon of gasoline increases an average of 2.9 cents.
However, because there are 42 gallons per barrel, we would expect a dollar increase
in a barrel of oil to yield a 2.4† cents per gallon increase (calculated as $1.00/42). It
does appear that the oil companies are taking some small advantage by adding an
extra half-cent per gallon. The coefficient of determination is .929, which indicates
that 92.9% of the variation in gasoline prices is explained by the variation in oil
prices.
†This
is a simplification. In fact a barrel of oil yields a variety of other profitable products. See
Exercise 2.14.
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153
EXERCISES
The following exercises require a computer and statistical
software.
4.90 Xr03-23 Refer to Exercise 3.23
a. Calculate the mean, median, and standard deviation of the scores of those who repaid and of
those who defaulted.
b. Do these statistics produce more useful information than the histograms?
4.91 Xr03-24 Refer to Exercise 3.24.
a. Draw box plots of the scores of those who repaid
and of those who defaulted.
b. Compare the information gleaned from the histograms to that contained in the box plots.
Which are better?
4.94 Xr03-56 Compute the coefficient of determination
and the least squares line for Exercise 3.56.
Compare this information with that developed by
the scatter diagram alone.
4.95 Xr03-59 Refer to Exercise 3.59. Calculate the coef-
ficient of determination and the least squares line.
Is this more informative than the scatter
diagram?
4.96 Xm03-07 a. Calculate the coefficients of the least
squares line for the data in Example 3.7.
b. Interpret the coefficients.
c. Is this information more useful than the information extracted from the scatter diagram?
4.92 Xr03-50 Calculate the coefficient of determination
for Exercise 3.50. Is this more informative than the
scatter diagram?
4.93 Xr03-51 Refer to Exercise 3.51. Compute the coeffi-
cients of the least squares line and compare your
results with the scatter diagram.
4 . 8 GENER AL GUIDELINES
F OR
4.97 Xr04-53 In Exercise 4.53, you drew box plots. Draw
histograms instead and compare the results.
4.98 Xr04-55 Refer to Exercise 4.55. Draw histograms of
the data. What have you learned?
E X PLOR I N G D ATA
The purpose of applying graphical and numerical techniques is to describe and summarize data. Statisticians usually apply graphical techniques as a first step because we need
to know the shape of the distribution. The shape of the distribution helps answer the
following questions:
1.
2.
3.
4.
Where is the approximate center of the distribution?
Are the observations close to one another, or are they widely dispersed?
Is the distribution unimodal, bimodal, or multimodal? If there is more than one
mode, where are the peaks, and where are the valleys?
Is the distribution symmetric? If not, is it skewed? If symmetric, is it bell shaped?
Histograms and box plots provide most of the answers. We can frequently make
several inferences about the nature of the data from the shape. For example, we can
assess the relative risk of investments by noting their spreads. We can attempt to
improve the teaching of a course by examining whether the distribution of final grades
is bimodal or skewed.
The shape can also provide some guidance on which numerical techniques to use.
As we noted in this chapter, the central location of highly skewed data may be more
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appropriately measured by the median. We may also choose to use the interquartile
range instead of the standard deviation to describe the spread of skewed data.
When we have an understanding of the structure of the data, we may do additional
analysis. For example, we often want to determine how one variable, or several variables,
affects another. Scatter diagrams, covariance, and the coefficient of correlation are useful
techniques for detecting relationships between variables. A number of techniques to be
introduced later in this book will help uncover the nature of these associations.
CHAPTER SUM M ARY
This chapter extended our discussion of descriptive statistics, which deals with methods of summarizing and presenting the essential information contained in a set of data. After
constructing a frequency distribution to obtain a general
idea about the distribution of a data set, we can use numerical measures to describe the central location and variability
of interval data. Three popular measures of central location,
or averages, are the mean, the median, and the mode. Taken
by themselves, these measures provide an inadequate
description of the data because they say nothing about the
extent to which the data vary. Information regarding the
variability of interval data is conveyed by such numerical
measures as the range, variance, and standard deviation.
For the special case in which a sample of measurements has a mound-shaped distribution, the Empirical Rule
provides a good approximation of the percentages of measurements that fall within one, two, and three standard
deviations of the mean. Chebysheff’s Theorem applies to all
sets of data no matter the shape of the histogram.
Measures of relative standing that were presented in
this chapter are percentiles and quartiles. The box plot
graphically depicts these measures as well as several others. The linear relationship between two interval variables
is measured by the covariance, the coefficient of correlation, the coefficient of determination, and the least
squares line.
Mean absolute deviation 110
Empirical Rule 113
Chebysheff’s Theorem 114
Skewed 114
Coefficient of variation 115
Percentiles 117
Quartiles 118
Interquartile range 120
Box plots 120
Outlier 121
Covariance 127
Coefficient of correlation 128
Least squares method 132
IMPORTANT TERMS
Measures of central location 98
Mean 98
Median 100
Mode 101
Modal class 102
Geometric mean 105
Measures of variability 108
Range 108
Variance 108
Standard deviation 108
Deviation 109
SYMBOLS
Symbol
Pronounced
Represents
m
s2
s
r
mu
Population mean
sigma squared
sigma
rho
Population variance
Population standard deviation
Population coefficient of correlation
a
Sum of
Summation
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NUMERICAL DESCRIPTIVE TECHNIQUES
Symbol
Pronounced
Represents
a xi
Sum of xi from 1 to n
Summation of n numbers
yN
b0
b1
y hat
Fitted or calculated value of y
b zero
y-Intercept
b one
Slope coefficient
155
n
i=1
FORMULAS
Population mean
N
Population covariance
N
a xi
a (xi - mx)(yi - my)
i=1
m =
sxy =
N
Sample mean
n
a (xi - x)(yi - y)
i=1
r =
Population variance
N
2
a (xi - m)
sxy
sxsy
Sample coefficient of correlation
i=1
r =
N
Sample variance
n
sxy
sxsy
Coefficient of determination
2
a (xi - x)
R2 = r2
i=1
Slope coefficient
n - 1
b1 =
Population standard deviation
s =
n - 1
Population coefficient of correlation
Largest observation – Smallest observation
s2 =
i=1
sxy =
n
Range
s2 =
N
Sample covariance
n
a xi
x =
i=1
2s2
sxy
s2x
y-intercept
Sample standard deviation
b0 = y - b1x
s = 2s2
COMPUTER OUTPUT AND INSTRUCTIONS
Technique
Mean
Median
Mode
Variance
Standard deviation
Descriptive statistics
Box plot
Least squares line
Excel
100
101
102
111
112
119
121
135
Minitab
100
101
102
111
113
120
122
136
(Continued)
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Technique
Covariance
Correlation
Coefficient of determination
Excel
Minitab
137
137
139
138
138
139
CHAPTER EXERCISES
4.99 Xr04-99* Osteoporosis is a condition in which bone
density decreases, often resulting in broken bones.
Bone density usually peaks at age 30 and decreases
thereafter. To understand more about the condition,
researchers recruited a random sample of women
aged 50 and older. Each woman’s bone density loss
was recorded.
a. Compute the mean and median of these data.
b. Compute the standard deviation of the bone density losses.
c. Describe what you have learned from the statistics.
4.100 Xr04-100* The temperature in December in Buffalo,
d. Discuss the information obtained here and in
Exercise 4.100. Which is more useful to the
manager?
4.103 Xr04-103* Chris Golfnut loves the game of golf. Chris
also loves statistics. Combining both passions, Chris
records a sample of 100 scores.
a. What statistics should Chris compute to describe
the scores?
b. Calculate the mean and standard deviation of the
scores.
c. Briefly describe what the statistics computed in
part (b) divulge.
New York, is often below 40 degrees Fahrenheit
(4 degrees Celsius). Not surprisingly, when the
National Football League Buffalo Bills play at home
in December, hot coffee is a popular item at the concession stand. The concession manager would like
to acquire more information so that he can manage
inventories more efficiently. The number of cups of
coffee sold during 50 games played in December in
Buffalo were recorded.
a. Determine the mean and median.
b. Determine the variance and standard deviation.
c. Draw a box plot.
d. Briefly describe what you have learned from your
statistical analysis.
4.104 Xr04-104* The Internet is growing rapidly with an
4.101 Refer to Exercise 4.99. In addition to the bone den-
recorded the number of putts as well as his scores.
Conduct an analysis of both sets of data. What conclusions can be achieved from the statistics?
sity losses, the ages of the women were also
recorded. Compute the coefficient of determination
and describe what this statistic tells you.
4.102 Refer to Exercise 4.100. Suppose that in addition to
recording the coffee sales, the manager also
recorded the average temperature (measured in
degrees Fahrenheit) during the game. These data
together with the number of cups of coffee sold were
recorded.
a. Compute the coefficient of determination.
b. Determine the coefficients of the least squares
line.
c. What have you learned from the statistics calculated in parts (a) and (b) about the relationship
between the number of cups of coffee sold and
the temperature?
increasing number of regular users. However,
among people older than 50, Internet use is still relatively low. To learn more about this issue, a sample
of 250 men and women older than 50 who had used
the Internet at least once were selected. The number
of hours on the Internet during the past month was
recorded.
a. Calculate the mean and median.
b. Calculate the variance and standard deviation.
c. Draw a box plot.
d. Briefly describe what you have learned from the
statistics you calculated.
4.105 Refer to Exercise 4.103. For each score, Chris also
4.106 Refer to Exercise 4.104. In addition to Internet use,
the numbers of years of education were recorded.
a. Compute the coefficient of determination.
b. Determine the coefficients of the least squares
line.
c. Describe what these statistics tell you about the
relationship between Internet use and education.
d. Discuss the information obtained here and in
Exercise 4.104.
4.107 Xr04-107* A sample was drawn of one-acre plots of
land planted with corn. The crop yields were
recorded. Calculate the descriptive statistics you
judge to be useful. Interpret these statistics.
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NUMERICAL DESCRIPTIVE TECHNIQUES
c. Describe what these statistics tell you about the
relationship between crop yield and the amount
of fertilizer.
d. Discuss the information obtained here and in
Exercise 4.107.
4.108 Refer to Exercise 4.107. For each plot, the amounts
of rainfall were also recorded.
a. Compute the coefficient of determination.
b. Determine the coefficients of the least squares
line.
c. Describe what these statistics tell you about the
relationship between crop yield and rainfall.
d. Discuss the information obtained here and in
Exercise 4.107.
4.110 Xr04-110 Increasing tuition has resulted in some stu-
4.109 Refer to Exercise 4.107. For each plot, the amounts
of fertilizer were recorded.
a. Compute the coefficient of determination.
b. Determine the coefficients of the least squares
line.
CASE 4.1
DATA
C04-01a
C04-01b
N
niques that allow us to conduct
more precise analyses we’ll
return to Case 3.1. Recall that there are
two issues in this discussion. First, is there
global warming; second, if so, is carbon
chapter to answer the same questions.
Column 2: Temperature anomalies
produced by the National
Here are the data sets you can work with.
C04-01a: Column 1: Months
Climatic Data Center
a.
numbered 1 to 1559
produced by the National
at the end of Chapter 3 were graphical
Climatic Data Center
C04-01b: Column 1: Monthly
Use the least squares method to
estimate average monthly changes
Column 2: Temperature anomalies
dioxide the cause? The only tools available
techniques including line charts and scat-
DATA
C04-02a
C04-02b
C04-02c
C04-02d
dents being saddled with large debts at graduation.
To examine this issue, a random sample of recent
graduates was asked to report whether they had student loans, and, if so, how much was the debt at
graduation.
a. Compute all three measures of central location.
b. What do these statistics reveal about student loan
debt at graduation?
Return to the Global Warming Question
ow that we have presented tech-
in temperature anomalies.
b.
Calculate the least squares line and
the coefficient of correlation
between CO2 levels and tempera-
ter diagrams. You are now invited to
carbon dioxide levels measured
ture anomalies and describe your
apply the more precise techniques in this
by the Mauna Loa Observatory
findings.
CASE 4.2
D
157
Another Return to the Global Warming Question
id you conclude in Case 4.1
another look at the same data. C04-02a
1997, and C04-02d contains the data
that Earth has warmed since
lists the temperature anomalies from
from 1998 to 2009. For each set of
1880 and that there is some
1880 to 1940, C04-02b lists the data
data, calculate the least squares line
linear relationship between CO2 and
from 1941 to 1975, C04-02c stores
and the coefficient of determination.
temperature anomalies? If so, here is
temperature anomalies from 1976 to
Report your findings.
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CHAPTER 4
CASE 4.3
The Effect of the Players’ Strike in the 2004-05 Hockey Season
he 2004–2005 hockey season was
T
of most players; as a result, the players
canceled because of a players’
association fought against it. The team
strike. The key issue in the labor
owners prevailed, and the collective bar-
dispute was a “salary cap.” The team
gaining agreement specified a salary
owners wanted a salary cap to cut their
cap of $39 million and a floor of $21.5
costs. The owners of small-market
million for the 2005–2006 season.
teams wanted the cap to help their
teams become competitive. Of course,
caps on salaries would lower the salaries
CASE 4.4
Conduct an analysis of the 2003–2004
season (C04-02b). For each season:
a.
Estimate how much on average a
DATA
C04-03a
C04-03b
team needs to spend to win one
more game.
b.
Measure the strength of the linear
relationship.
c.
season (C04-02a) and the 2005–2006
Discuss the differences between the
two seasons.
Quebec Referendum Vote: Was
There Electoral Fraud?*
© Bettman/Corbis
CH004.qxd
ince the 1960s, Quebecois (citi-
S
appointed by the proseparatist provin-
zens of the province of Quebec)
cial government, rejected a dispropor-
have been debating whether to
tionate number of ballots in ridings
separate from Canada and form an
(electoral districts) where the percent-
high percentages of rejected ballots, this
independent nation. A referendum was
age of yes votes was low and where
too would constitute evidence of fraud
held on October 30, 1995, in which the
there are large numbers of Allophone
on the part of the scrutineers and possi-
people of Quebec voted not to separate.
(people whose first language is neither
bly the government.
The vote was extremely close with the
English nor French) and Anglophone
To determine the veracity of the charges,
“no” side winning by only 52,448 votes.
(English-speaking) residents. (Electoral
the following variables were recorded for
A large number of no votes was cast by
laws require the rejection of ballots that
each riding.
the non-Francophone (non–French
do not appear to be properly marked.)
speaking) people of Quebec, who make
They were outraged that in a strong
Percentage of rejected ballots in
up about 20% of the population and
democracy like Canada, votes would be
referendum
who very much want to remain
rigged much as they are in many non-
Percentage of “yes” votes
Canadians. The remaining 80% are
democratic countries around the world.
Francophones, a majority of whom
voted “yes.”
If, in ridings where there was a low percentage of “yes” votes, there was a high
DATA
C04-04
Percentage of Allophones
Percentage of Anglophones
After the votes were counted, it became
percentage of rejected ballots, this
Conduct a statistical analysis of these
clear that the tallied vote was much
would be evidence of electoral fraud.
data to determine whether there are
closer than it should have been.
Moreover, if, in ridings where there were
indications that electoral fraud took
Supporters of the no side charged that
large percentages of Allophone or
place.
poll scrutineers, all of whom were
Anglophone voters (or both), there were
*This case is based on “Voting Irregularities in the 1995 Referendum on Quebec Sovereignty” by Jason Cawley and Paul Sommers, Chance, Vol. 9, No. 4, Fall, 1996. We
are grateful to Dr. Paul Sommers, Middlebury College, for his assistance in writing this case.
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NUMERICAL DESCRIPTIVE TECHNIQUES
A P P E N D I X 4 REVIEW
OF
159
DESCRIPTIVE TECHNIQUES
Here is a list of the statistical techniques introduced in Chapters 2, 3, and 4. This is followed by a flowchart designed to help you select the most appropriate method to use to
address any problem requiring a descriptive method.
To provide practice in identifying the correct descriptive method to use we have
created a number of review exercises. These are in Keller’s website Appendix
Descriptive Techniques Review Exercises.
Graphical Techniques
Histogram
Stem-and-leaf display
Ogive
Bar chart
Pie chart
Scatter diagram
Line chart (time series)
Box plot
Numerical Techniques
Measures of Central Location
Mean
Median
Mode
Geometric mean (growth rates)
Measures of Variability
Range
Variance
Standard deviation
Coefficient of variation
Interquartile range
Measures of Relative Standing
Percentiles
Quartiles
Measures of Linear Relationship
Covariance
Coefficient of correlation
Coefficient of determination
Least squares line
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Flowchart: Graphical and Numerical Techniques
Problem objective?
Describe a set of data
Describe relationship between two variables
Data type?
Data type?
Interval
Ordinal
Nominal
Nominal
Ordinal
Interval
Graphical
Graphical
Graphical
Graphical
Histogram
Stem-and-leaf
Ogive
Treat as
nominal
Bar chart
Pie chart
Numerical
Numerical
Box plot
Line chart*
Mode
Descriptive measure?
Central location
Variability
Relative standing
Median
Interquartile range
Percentiles
Quartiles
Scatter diagram
Numerical
Covariance
Correlation
Determination
Least squares line
Graphical
Graphical
Treat as
nominal
Bar chart of
a crossclassification
table
Numerical
Descriptive measure?
Central location
Variability
Relative standing
Mean
Range
Variance
Standard deviation
Coefficient of variation
Percentiles
Quartiles
Median
Mode
Interquartile range
Geometric mean†
*Time-series data
†Growth rates
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5
© Robert Hardholt/Shutterstock
DATA COLLECTION
AND SAMPLING
5.1
Methods of Collecting Data
5.2
Sampling
5.3
Sampling Plans
5.4
Sampling and Nonsampling Errors
The census, which is conducted every 10 years in the United States, serves an important function. It
is the basis for deciding how many congressional representatives and how many votes in the electoral college each state will have. Businesses often use the information derived from the census to
help make decisions about products, advertising, and plant locations.
One of the problems with the census is the issue of undercounting, which occurs when
Courtesy, US Census Bureau
Sampling and the Census
some people are not included. For example, the 1990 census reported that 12.05% of adults were
African American; the true value was 12.41%. To address undercounting, the Census Bureau
adjusts the numbers it gets from the census. The adjustment is based on another survey. The
mechanism is called the Accuracy and Coverage Evaluation. Using sampling methods described
161
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in this chapter, the Census Bureau is able to adjust the numbers in American subgroups. For example, the Bureau may discover that the number of Hispanics has been undercounted or that the number of people living in California has not been
accurately counted.
Later in this chapter we’ll discuss how the sampling is conducted and how the adjustments are made.
I N T RO D U C T I O N
5 . 1 METHODS
I
n Chapter 1, we briefly introduced the concept of statistical inference—the process
of inferring information about a population from a sample. Because information
about populations can usually be described by parameters, the statistical technique
used generally deals with drawing inferences about population parameters from sample
statistics. (Recall that a parameter is a measurement about a population, and a statistic is
a measurement about a sample.)
Working within the covers of a statistics textbook, we can assume that population
parameters are known. In real life, however, calculating parameters is virtually impossible because populations tend to be very large. As a result, most population parameters
are not only unknown but also unknowable. The problem that motivates the subject of
statistical inference is that we often need information about the value of parameters in
order to make decisions. For example, to make decisions about whether to expand a line
of clothing, we may need to know the mean annual expenditure on clothing by North
American adults. Because the size of this population is approximately 200 million,
determining the mean is prohibitive. However, if we are willing to accept less than
100% accuracy, we can use statistical inference to obtain an estimate. Rather than investigating the entire population, we select a sample of people, determine the annual
expenditures on clothing in this group, and calculate the sample mean. Although the
probability that the sample mean will equal the population mean is very small, we would
expect them to be close. For many decisions, we need to know how close. We postpone
that discussion until Chapters 10 and 11. In this chapter, we will discuss the basic concepts and techniques of sampling itself. But first we take a look at various sources for
collecting data.
OF
C O L L E C T I N G D ATA
Most of this book addresses the problem of converting data into information. The
question arises, where do data come from? The answer is that a large number of methods produce data. Before we proceed however, we’ll remind you of the definition of
data introduced in Section 2.1. Data are the observed values of a variable; that is, we
define a variable or variables that are of interest to us and then proceed to collect observations of those variables.
Direct Observation
The simplest method of obtaining data is by direct observation. When data are gathered in this way, they are said to be observational. For example, suppose that a
researcher for a pharmaceutical company wants to determine whether aspirin actually
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DATA COLLECTION AND SAMPLING
163
reduces the incidence of heart attacks. Observational data may be gathered by selecting
a sample of men and women and asking each whether he or she has taken aspirin regularly over the past 2 years. Each person would be asked whether he or she had suffered
a heart attack over the same period. The proportions reporting heart attacks would be
compared and a statistical technique that is introduced in Chapter 13 would be used to
determine whether aspirin is effective in reducing the likelihood of heart attacks. There
are many drawbacks to this method. One of the most critical is that it is difficult to produce useful information in this way. For example, if the statistics practitioner concludes
that people who take aspirin suffer fewer heart attacks, can we conclude that aspirin is
effective? It may be that people who take aspirin tend to be more health conscious, and
health-conscious people tend to have fewer heart attacks. The one advantage to direct
observation is that it is relatively inexpensive.
Experiments
A more expensive but better way to produce data is through experiments. Data produced in this manner are called experimental. In the aspirin illustration, a statistics
practitioner can randomly select men and women. The sample would be divided into
two groups. One group would take aspirin regularly, and the other would not. After
2 years, the statistics practitioner would determine the proportion of people in each
group who had suffered heart attacks, and statistical methods again would be used to
determine whether aspirin works. If we find that the aspirin group suffered fewer heart
attacks, then we may more confidently conclude that taking aspirin regularly is a
healthy decision.
Surveys
One of the most familiar methods of collecting data is the survey, which solicits information from people concerning such things as their income, family size, and opinions
on various issues. We’re all familiar, for example, with opinion polls that accompany
each political election. The Gallup Poll and the Harris Survey are two well-known surveys of public opinion whose results are often reported by the media. But the majority
of surveys are conducted for private use. Private surveys are used extensively by market
researchers to determine the preferences and attitudes of consumers and voters. The
results can be used for a variety of purposes, from helping to determine the target market for an advertising campaign to modifying a candidate’s platform in an election campaign. As an illustration, consider a television network that has hired a market research
firm to provide the network with a profile of owners of luxury automobiles, including
what they watch on television and at what times. The network could then use this information to develop a package of recommended time slots for Cadillac commercials,
including costs, which it would present to General Motors. It is quite likely that many
students reading this book will one day be marketing executives who will “live and die”
by such market research data.
An important aspect of surveys is the response rate. The response rate is the proportion of all people who were selected who complete the survey. As we discuss in the
next section, a low response rate can destroy the validity of any conclusion resulting
from the statistical analysis. Statistics practitioners need to ensure that data are reliable.
Personal Interview Many researchers feel that the best way to survey people is by
means of a personal interview, which involves an interviewer soliciting information
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from a respondent by asking prepared questions. A personal interview has the
advantage of having a higher expected response rate than other methods of data collection. In addition, there will probably be fewer incorrect responses resulting from
respondents misunderstanding some questions because the interviewer can clarify misunderstandings when asked to. But the interviewer must also be careful not to say too
much for fear of biasing the response. To avoid introducing such biases, as well as to
reap the potential benefits of a personal interview, the interviewer must be well trained
in proper interviewing techniques and well informed on the purpose of the study. The
main disadvantage of personal interviews is that they are expensive, especially when
travel is involved.
Telephone Interview A telephone interview is usually less expensive, but it is also
less personal and has a lower expected response rate. Unless the issue is of interest,
many people will refuse to respond to telephone surveys. This problem is exacerbated
by telemarketers trying to sell something.
Self-Administered Survey A third popular method of data collection is the selfadministered questionnaire, which is usually mailed to a sample of people. This is an
inexpensive method of conducting a survey and is therefore attractive when the number
of people to be surveyed is large. But self-administered questionnaires usually have a
low response rate and may have a relatively high number of incorrect responses due to
respondents misunderstanding some questions.
Questionnaire Design Whether a questionnaire is self-administered or completed
by an interviewer, it must be well designed. Proper questionnaire design takes knowledge, experience, time, and money. Some basic points to consider regarding questionnaire design follow.
1.
First and foremost, the questionnaire should be kept as short as possible to
encourage respondents to complete it. Most people are unwilling to spend much
time filling out a questionnaire.
2.
The questions themselves should also be short, as well as simply and clearly
worded, to enable respondents to answer quickly, correctly, and without ambiguity. Even familiar terms such as “unemployed” and “family” must be defined carefully because several interpretations are possible.
3.
4.
Questionnaires often begin with simple demographic questions to help respondents get started and become comfortable quickly.
5.
6.
Open-ended questions provide an opportunity for respondents to express opinions
more fully, but they are time consuming and more difficult to tabulate and analyze.
Dichotomous questions (questions with only two possible responses such as “yes”
and “no” and multiple-choice questions) are useful and popular because of their
simplicity, but they also have possible shortcomings. For example, a respondent’s
choice of yes or no to a question may depend on certain assumptions not stated in
the question. In the case of a multiple-choice question, a respondent may feel that
none of the choices offered is suitable.
Avoid using leading questions, such as “Wouldn’t you agree that the statistics
exam was too difficult?” These types of questions tend to lead the respondent to a
particular answer.
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DATA COLLECTION AND SAMPLING
7.
8.
165
Time permitting, it is useful to pretest a questionnaire on a small number of people in order to uncover potential problems such as ambiguous wording.
Finally, when preparing the questions, think about how you intend to tabulate
and analyze the responses. First, determine whether you are soliciting values (i.e.,
responses) for an interval variable or a nominal variable. Then consider which
type of statistical techniques—descriptive or inferential—you intend to apply to
the data to be collected, and note the requirements of the specific techniques to
be used. Thinking about these questions will help ensure that the questionnaire is
designed to collect the data you need.
Whatever method is used to collect primary data, we need to know something
about sampling, the subject of the next section.
EXERCISES
5.1 Briefly describe the difference between observa-
tional and experimental data.
5.2 A soft drink manufacturer has been supplying its
cola drink in bottles to grocery stores and in cans to
small convenience stores. The company is analyzing
sales of this cola drink to determine which type of
packaging is preferred by consumers.
a. Is this study observational or experimental?
Explain your answer.
b. Outline a better method for determining whether
a store will be supplied with cola in bottles or in
cans so that future sales data will be more helpful
in assessing the preferred type of packaging.
5.3 a. Briefly describe how you might design a study to
investigate the relationship between smoking and
lung cancer.
b. Is your study in part (a) observational or experimental? Explain why.
5.4 a. List three methods of conducting a survey of
people.
b. Give an important advantage and disadvantage of
each of the methods listed in part (a).
5.5 List five important points to consider when design-
ing a questionnaire.
5 . 2 S A M PL I N G
The chief motive for examining a sample rather than a population is cost. Statistical
inference permits us to draw conclusions about a population parameter based on a sample that is quite small in comparison to the size of the population. For example, television
executives want to know the proportion of television viewers who watch a network’s programs. Because 100 million people may be watching television in the United States on a
given evening, determining the actual proportion of the population that is watching certain programs is impractical and prohibitively expensive. The Nielsen ratings provide
approximations of the desired information by observing what is watched by a sample of
5,000 television viewers. The proportion of households watching a particular program
can be calculated for the households in the Nielsen sample. This sample proportion is
then used as an estimate of the proportion of all households (the population proportion)
that watched the program.
Another illustration of sampling can be taken from the field of quality management. To ensure that a production process is operating properly, the operations manager needs to know what proportion of items being produced is defective. If the quality
technician must destroy the item to determine whether it is defective, then there is no
alternative to sampling: A complete inspection of the product population would destroy
the entire output of the production process.
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We know that the sample proportion of television viewers or of defective items is
probably not exactly equal to the population proportion we want to estimate.
Nonetheless, the sample statistic can come quite close to the parameter it is designed to
estimate if the target population (the population about which we want to draw inferences) and the sampled population (the actual population from which the sample has
been taken) are the same. In practice, these may not be the same. One of statistics’ most
famous failures illustrates this phenomenon.
The Literary Digest was a popular magazine of the 1920s and 1930s that had correctly
predicted the outcomes of several presidential elections. In 1936, the Digest predicted that
the Republican candidate, Alfred Landon, would defeat the Democratic incumbent,
Franklin D. Roosevelt, by a 3 to 2 margin. But in that election, Roosevelt defeated
Landon in a landslide victory, garnering the support of 62% of the electorate. The source
of this blunder was the sampling procedure, and there were two distinct mistakes.* First,
the Digest sent out 10 million sample ballots to prospective voters. However, most of the
names of these people were taken from the Digest’s subscription list and from telephone
directories. Subscribers to the magazine and people who owned telephones tended to be
wealthier than average and such people then, as today, tended to vote Republican. In addition, only 2.3 million ballots were returned resulting in a self-selected sample.
Self-selected samples are almost always biased because the individuals who participate in them are more keenly interested in the issue than are the other members of
the population. You often find similar surveys conducted today when radio and television stations ask people to call and give their opinion on an issue of interest. Again, only
listeners who are concerned about the topic and have enough patience to get through to
the station will be included in the sample. Hence, the sampled population is composed
entirely of people who are interested in the issue, whereas the target population is made
up of all the people within the listening radius of the radio station. As a result, the conclusions drawn from such surveys are frequently wrong.
An excellent example of this phenomenon occurred on ABC’s Nightline in 1984.
Viewers were given a 900 telephone number (cost: 50 cents) and asked to phone in their
responses to the question of whether the United Nations should continue to be located
in the United States. More than 186,000 people called, with 67% responding “no.” At
the same time, a (more scientific) market research poll of 500 people revealed that 72%
wanted the United Nations to remain in the United States. In general, because the true
value of the parameter being estimated is never known, these surveys give the impression of providing useful information. In fact, the results of such surveys are likely to be
no more accurate than the results of the 1936 Literary Digest poll or Nightline’s phonein show. Statisticians have coined two terms to describe these polls: SLOP (self-selected
opinion poll) and Oy vey (from the Yiddish lament), both of which convey the contempt
that statisticians have for such data-gathering processes.
* Many statisticians ascribe the Literary Digest’s statistical debacle to the wrong causes. For an understanding of what really happened, read Maurice C. Bryson, “The Literary Digest Poll: Making of a
Statistical Myth” American Statistician 30(4) (November 1976): 184–185.
EXERCISES
5.6 For each of the following sampling plans, indicate
why the target population and the sampled population are not the same.
a. To determine the opinions and attitudes of customers who regularly shop at a particular mall, a
surveyor stands outside a large department store
in the mall and randomly selects people to participate in the survey.
b. A library wants to estimate the proportion of its
books that have been damaged. The librarians
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decide to select one book per shelf as a sample by
measuring 12 inches from the left edge of each
shelf and selecting the book in that location.
c. Political surveyors visit 200 residences during
one afternoon to ask eligible voters present in the
house at the time whom they intend to vote for.
5.9 A regular feature in a newspaper asks readers to
5.7 a. Describe why the Literary Digest poll of 1936 has
5.10 Suppose your statistics professor distributes a
become infamous.
b. What caused this poll to be so wrong?
5.8 a. What is meant by self-selected sample?
b. Give an example of a recent poll that involved a
self-selected sample.
c. Why are self-selected samples not desirable?
respond via e-mail to a survey that requires a yes
or no response. In the following day’s newspaper,
the percentage of yes and no responses are
reported. Discuss why we should ignore these
statistics.
questionnaire about the course. One of the
questions asks, “Would you recommend this course
to a friend?” Can the professor use the results
to infer something about all statistics courses?
Explain.
5 . 3 S A M PL I N G P L A N S
Our objective in this section is to introduce three different sampling plans: simple
random sampling, stratified random sampling, and cluster sampling. We begin our presentation with the most basic design.
Simple Random Sampling
Simple Random Sample
A simple random sample is a sample selected in such a way that every possible sample with the same number of observations is equally likely to be
chosen.
One way to conduct a simple random sample is to assign a number to each element in
the population, write these numbers on individual slips of paper, toss them into a hat,
and draw the required number of slips (the sample size, n) from the hat. This is the kind
of procedure that occurs in raffles, when all the ticket stubs go into a large rotating
drum from which the winners are selected.
Sometimes the elements of the population are already numbered. For example, virtually all adults have Social Security numbers (in the United States) or Social Insurance
numbers (in Canada); all employees of large corporations have employee numbers;
many people have driver’s license numbers, medical plan numbers, student numbers,
and so on. In such cases, choosing which sampling procedure to use is simply a matter
of deciding how to select from among these numbers.
In other cases, the existing form of numbering has built-in flaws that make it inappropriate as a source of samples. Not everyone has a phone number, for example, so the
telephone book does not list all the people in a given area. Many households have two
(or more) adults but only one phone listing. Couples often list the phone number under
the man’s name, so telephone listings are likely to be disproportionately male. Some
people do not have phones, some have unlisted phone numbers, and some have more
than one phone; these differences mean that each element of the population does not
have an equal probability of being selected.
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After each element of the chosen population has been assigned a unique number,
sample numbers can be selected at random. A random number table can be used to
select these sample numbers. (See, for example, CRC Standard Management Tables,
W. H. Beyer, ed., Boca Raton FL: CRC Press.) Alternatively, we can use Excel to perform this function.
EXAMPLE
5.1
Random Sample of Income Tax Returns
A government income tax auditor has been given responsibility for 1,000 tax returns.
A computer is used to check the arithmetic of each return. However, to determine
whether the returns have been completed honestly, the auditor must check each entry
and confirm its veracity. Because it takes, on average, 1 hour to completely audit a
return and she has only 1 week to complete the task, the auditor has decided to randomly select 40 returns. The returns are numbered from 1 to 1,000. Use a computer
random-number generator to select the sample for the auditor.
SOLUTION
We generated 50 numbers between 1 and 1,000 even though we needed only 40 numbers. We did so because it is likely that there will be some duplicates. We will use the first
40 unique random numbers to select our sample. The following numbers were generated by Excel. The instructions for both Excel and Minitab are provided here. [Notice
that the 24th and 36th (counting down the columns) numbers generated were the
same—467.]
Computer-Generated Random Numbers
383
101
597
900
885
959
15
408
864
139
246
46
33
165
220
18
286
344
554
358
372
356
911
467
427
304
976
807
992
257
952
54
706
817
973
467
301
751
352
776
75
199
65
359
488
512
374
986
41
231
EXCEL
INSTRUCTIONS
1. Click Data, Data Analysis, and Random Number Generation.
2. Specify the Number of Variables (1) and the Number of Random Numbers (50).
3. Select Uniform Distribution.
4. Specify the range of the uniform distribution (Parameters) (0 and 1).
5. Click OK. Column A will fill with 50 numbers that range between 0 and 1.
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6. Multiply column A by 1,000 and store the products in column B.
7. Make cell C1 active, and click fx, Math & Trig, ROUNDUP, and OK.
8. Specify the first number to be rounded (B1).
9. Type the number of digits (decimal places) (0). Click OK.
10. Complete column C.
The first five steps command Excel to generate 50 uniformly distributed random numbers between 0 and 1 to be stored in column A. Steps 6 through 10 convert these random
numbers to integers between 1 and 1,000. Each tax return has the same probability
(1/1,000 ⫽ .001) of being selected. Thus, each member of the population is equally likely
to be included in the sample.
MINITAB
INSTRUCTIONS
1. Click Calc, Random Data, and Integer . . ..
2. Type the number of random numbers you wish (50).
3. Specify where the numbers are to be stored (C1).
4. Specify the Minimum value (1).
5. Specify the Maximum value (1000). Click OK.
INTERPRET
The auditor would examine the tax returns selected by the computer. She would pick
returns numbered 383, 101, 597, . . . , 352, 776, and 75 (the first 40 unique numbers).
Each of these returns would be audited to determine whether it is fraudulent. If the
objective is to audit these 40 returns, no statistical procedure would be employed.
However, if the objective is to estimate the proportion of all 1,000 returns that are dishonest, then she would use one of the inferential techniques presented later in this book.
Stratified Random Sampling
In making inferences about a population, we attempt to extract as much information as
possible from a sample. The basic sampling plan, simple random sampling, often accomplishes this goal at low cost. Other methods, however, can be used to increase the amount
of information about the population. One such procedure is stratified random sampling.
Stratified Random Sample
A stratified random sample is obtained by separating the population into
mutually exclusive sets, or strata, and then drawing simple random samples
from each stratum.
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Examples of criteria for separating a population into strata (and of the strata themselves) follow.
1. Gender
male
female
2. Age
under 20
20–30
31–40
41–50
51–60
over 60
3. Occupation
professional
clerical
blue-collar
other
4. Household income
under $25,000
$25,000–$39,999
$40,000–$60,000
over $60,000
To illustrate, suppose a public opinion survey is to be conducted to determine how
many people favor a tax increase. A stratified random sample could be obtained by selecting a random sample of people from each of the four income groups we just described.
We usually stratify in a way that enables us to obtain particular kinds of information. In
this example, we would like to know whether people in the different income categories
differ in their opinions about the proposed tax increase, because the tax increase will
affect the strata differently. We avoid stratifying when there is no connection between
the survey and the strata. For example, little purpose is served in trying to determine
whether people within religious strata have divergent opinions about the tax increase.
One advantage of stratification is that, besides acquiring information about the
entire population, we can also make inferences within each stratum or compare strata.
For instance, we can estimate what proportion of the lowest income group favors the
tax increase, or we can compare the highest and lowest income groups to determine
whether they differ in their support of the tax increase.
Any stratification must be done in such a way that the strata are mutually exclusive:
Each member of the population must be assigned to exactly one stratum. After the population has been stratified in this way, we can use simple random sampling to generate
the complete sample. There are several ways to do this. For example, we can draw random samples from each of the four income groups according to their proportions in the
population. Thus, if in the population the relative frequencies of the four groups are as
listed here, our sample will be stratified in the same proportions. If a total sample of
1,000 is to be drawn, then we will randomly select 250 from stratum 1, 400 from stratum 2, 300 from stratum 3, and 50 from stratum 4.
Stratum
Income Categories ($)
Population Proportions (%)
1
2
3
4
Less than 25,000
25,000–39,999
40,000–60,000
More than 60,000
25
40
30
5
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The problem with this approach, however, is that if we want to make inferences
about the last stratum, a sample of 50 may be too small to produce useful information.
In such cases, we usually increase the sample size of the smallest stratum to ensure that
the sample data provide enough information for our purposes. An adjustment must
then be made before we attempt to draw inferences about the entire population. The
required procedure is beyond the level of this book. We recommend that anyone planning such a survey consult an expert statistician or a reference book on the subject.
Better still, become an expert statistician yourself by taking additional statistics courses.
Cluster Sampling
Cluster Sample
A cluster sample is a simple random sample of groups or clusters of elements.
Cluster sampling is particularly useful when it is difficult or costly to develop a complete list of the population members (making it difficult and costly to generate a simple
random sample). It is also useful whenever the population elements are widely dispersed
geographically. For example, suppose we wanted to estimate the average annual household income in a large city. To use simple random sampling, we would need a complete
list of households in the city from which to sample. To use stratified random sampling,
we would need the list of households, and we would also need to have each household
categorized by some other variable (such as age of household head) in order to develop
the strata. A less-expensive alternative would be to let each block within the city represent a cluster. A sample of clusters could then be randomly selected, and every household within these clusters could be questioned to determine income. By reducing the
distances the surveyor must cover to gather data, cluster sampling reduces the cost.
But cluster sampling also increases sampling error (see Section 5.4) because households belonging to the same cluster are likely to be similar in many respects, including
household income. This can be partially offset by using some of the cost savings to
choose a larger sample than would be used for a simple random sample.
Sample Size
Whichever type of sampling plan you select, you still have to decide what size sample to
use. Determining the appropriate sample size will be addressed in detail in Chapters 10
and 12. Until then, we can rely on our intuition, which tells us that the larger the sample
size is, the more accurate we can expect the sample estimates to be.
To adjust for undercounting, the Census Bureau conducts cluster sampling. The clusters are geographic
blocks. For the year 2000 census, the bureau randomly sampled 11,800 blocks, which contained 314,000
housing units. Each unit was intensively revisited to ensure that all residents were counted. From the
results of this survey, the Census Bureau estimated the number of people missed by the first census in
various subgroups, defined by several variables including gender, race, and age. Because of the impor-
Courtesy, US Census Bureau
Sampling and the Census
tance of determining state populations, adjustments were made to state totals. For example, by comparing the results of the census and of the sampling, the Bureau determined that the undercount in the
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state of Texas was 1.7087%. The official census produced a state population of 20,851,820. Taking 1.7087% of this total produced
an adjustment of 356,295. Using this method changed the population of the state of Texas to 21,208,115.
It should be noted that this process is contentious. The controversy concerns the way in which subgroups are defined.
Changing the definition alters the undercounts, making this statistical technique subject to politicking.
EXERCISES
5.11 A statistics practitioner would like to conduct a sur-
5.14 A telemarketing firm has recorded the households
vey to ask people their views on a proposed new
shopping mall in their community. According to the
latest census, there are 500 households in the community. The statistician has numbered each household (from 1 to 500), and she would like to randomly
select 25 of these households to participate in the
study. Use Excel or Minitab to generate the sample.
that have purchased one or more of the company’s
products. These number in the millions. The firm
would like to conduct a survey of purchasers to
acquire information about their attitude concerning
the timing of the telephone calls. The president of
the company would like to know the views of all purchasers but would also like to compare the attitudes
of people in the West, South, North, and East.
Describe a suitable sampling plan.
5.12 A safety expert wants to determine the proportion of
cars in his state with worn tire treads. The state
license plate contains six digits. Use Excel or Minitab
to generate a sample of 20 cars to be examined.
5.13 A large university campus has 60,000 students. The
president of the students’ association wants to conduct
a survey of the students to determine their views on an
increase in the student activity fee. She would like to
acquire information about all the students but would
also like to compare the school of business, the faculty
of arts and sciences, and the graduate school. Describe
a sampling plan that accomplishes these goals.
5 . 4 S A M PL I N G
AND
5.15 The operations manager of a large plant with four
departments wants to estimate the person-hours lost
per month from accidents. Describe a sampling plan
that would be suitable for estimating the plantwide
loss and for comparing departments.
5.16 A statistics practitioner wants to estimate the mean
age of children in his city. Unfortunately, he does
not have a complete list of households. Describe
a sampling plan that would be suitable for his
purposes.
N O N S A M PL I N G E R ROR S
Two major types of error can arise when a sample of observations is taken from a population: sampling error and nonsampling error. Anyone reviewing the results of sample surveys
and studies, as well as statistics practitioners conducting surveys and applying statistical
techniques, should understand the sources of these errors.
Sampling Error
Sampling error refers to differences between the sample and the population that exists
only because of the observations that happened to be selected for the sample. Sampling
error is an error that we expect to occur when we make a statement about a population
that is based only on the observations contained in a sample taken from the population.
To illustrate, suppose that we wish to determine the mean annual income of North
American blue-collar workers. To determine this parameter we would have to ask each
North American blue-collar worker what his or her income is and then calculate the mean
of all the responses. Because the size of this population is several million, the task is both
expensive and impractical. We can use statistical inference to estimate the mean income ␮
of the population if we are willing to accept less than 100% accuracy. We record the
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incomes of a sample of the workers and find the mean x of this sample of incomes. This
sample mean is an estimate, of the desired, population mean. But the value of the sample
mean will deviate from the population mean simply by chance because the value of
the sample mean depends on which incomes just happened to be selected for the sample. The
difference between the true (unknown) value of the population mean and its estimate, the
sample mean, is the sampling error. The size of this deviation may be large simply because of
bad luck—bad luck that a particularly unrepresentative sample happened to be selected. The
only way we can reduce the expected size of this error is to take a larger sample.
Given a fixed sample size, the best we can do is to state the probability that the
sampling error is less than a certain amount (as we will discuss in Chapter 10). It is common today for such a statement to accompany the results of an opinion poll. If an opinion poll states that, based on sample results, the incumbent candidate for mayor has the
support of 54% of eligible voters in an upcoming election, the statement may be
accompanied by the following explanatory note: “This percentage is correct to within
three percentage points, 19 times out of 20.” This statement means that we estimate
that the actual level of support for the candidate is between 51% and 57%, and that in
the long run this type of procedure is correct 95% of the time.
S E E I N G S TAT I S T I C S
applet 3 Sampling
When you select this applet, you will see
Show All button to see information for
100 circles. Imagine that each of the
all the households. How well did your
circles represents a household. You
sample estimate the true proportion?
want to estimate the proportion of
Click the Reset button to try again.
households having high-speed Internet
(Note: This page uses a randomly
access (DSL, cable modem, etc.). You
determined base proportion each time it
may collect data from a sample of
is loaded or reloaded.)
10 households by clicking on a
household’s circle. If the circle turns red,
the household has high-speed Internet
access. If the circle turns green, the
household does not have high-speed
access. After collecting your sample and
obtaining your estimate, click on the
Applet Exercises
3.1 Run the applet 25 times. How
many times did the sample
proportion equal the population
proportion of homes with high-
proportion?
speed Internet access as well as the
3.2 Run the applet 20 times. For each
simulation, record the sample
population proportion. Compute
the average sampling error.
Nonsampling Error
Nonsampling error is more serious than sampling error because taking a larger sample
won’t diminish the size, or the possibility of occurrence, of this error. Even a census can
(and probably will) contain nonsampling errors. Nonsampling errors result from mistakes
made in the acquisition of data or from the sample observations being selected improperly.
1.
Errors in data acquisition. This type of error arises from the recording of incorrect
responses. Incorrect responses may be the result of incorrect measurements being
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taken because of faulty equipment, mistakes made during transcription from primary sources, inaccurate recording of data because terms were misinterpreted, or
inaccurate responses were given to questions concerning sensitive issues such as
sexual activity or possible tax evasion.
2.
Nonresponse error. Nonresponse error refers to error (or bias) introduced when
responses are not obtained from some members of the sample. When this happens, the sample observations that are collected may not be representative of the
target population, resulting in biased results (as was discussed in Section 5.2).
Nonresponse can occur for a number of reasons. An interviewer may be unable to
contact a person listed in the sample, or the sampled person may refuse to
respond for some reason. In either case, responses are not obtained from a sampled person, and bias is introduced. The problem of nonresponse is even greater
when self-administered questionnaires are used rather than an interviewer, who
can attempt to reduce the nonresponse rate by means of callbacks. As noted previously, the Literary Digest fiasco was largely the result of a high nonresponse rate,
resulting in a biased, self-selected sample.
3.
Selection bias. Selection bias occurs when the sampling plan is such that some members of the target population cannot possibly be selected for inclusion in the sample.
Together with nonresponse error, selection bias played a role in the Literary Digest
poll being so wrong, as voters without telephones or without a subscription to
Literary Digest were excluded from possible inclusion in the sample taken.
EXERCISES
5.17 a. Explain the difference between sampling error
and nonsampling error.
b. Which type of error in part (a) is more serious?
Why?
5.18 Briefly describe three types of nonsampling error.
5.19 Is it possible for a sample to yield better results than
a census? Explain.
CHAPTER SUM M ARY
Because most populations are very large, it is extremely
costly and impractical to investigate each member of the
population to determine the values of the parameters. As a
practical alternative, we take a sample from the population
and use the sample statistics to draw inferences about the
parameters. Care must be taken to ensure that the sampled
population is the same as the target population.
We can choose from among several different sampling
plans, including simple random sampling, stratified random sampling, and cluster sampling. Whatever sampling
plan is used, it is important to realize that both sampling
error and nonsampling error will occur and to understand
what the sources of these errors are.
IMPORTANT TERMS
Observational 162
Experimental 163
Survey 163
Response rate 163
Estimate 165
Target population 166
Sampled population 166
Self-selected sample 166
Simple random sample 167
Stratified random sample 169
Cluster sample 171
Sampling error 172
Nonsampling error 173
Nonresponse error (bias) 174
Selection bias 174
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6
© Gary Buss/Taxi/Getty Images
6.1
Assigning Probability to Events
6.2
Joint, Marginal, and Conditional Probability
6.3
Probability Rules and Trees
6.4
Bayes’s Law
6.5
Identifying the Correct Method
Auditing Tax Returns
Government auditors routinely check tax returns to determine whether calculation errors were made.
They also attempt to detect fraudulent returns. There are several methods that dishonest taxpayers
use to evade income tax. One method is not to declare various sources of income. Auditors have
several detection methods, including spending patterns. Another form of tax fraud is to invent
deductions that are not real. After analyzing the returns of thousands of self-employed taxpayers, an
© Gary Buss/Taxi/Getty Images
PROBABILITY
auditor has determined that 45% of fraudulent returns contain two suspicious deductions, 28% contain one suspicious deduction, and the rest no suspicious deductions. Among honest returns the rates
are 11% for two deductions, 18% for one deduction, and 71% for no deductions. The auditor believes
See page 202 for the
answer.
that 5% of the returns of self-employed individuals contain significant fraud. The auditor has just
received a tax return for a self-employed individual that contains one suspicious expense deduction.
What is the probability that this tax return contains significant fraud?
175
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CHAPTER 6
I N T RO D U C T I O N
I
n Chapters 2, 3, and 4, we introduced graphical and numerical descriptive methods.
Although the methods are useful on their own, we are particularly interested in
developing statistical inference. As we pointed out in Chapter 1, statistical inference
is the process by which we acquire information about populations from samples. A critical component of inference is probability because it provides the link between the population and the sample.
Our primary objective in this and the following two chapters is to develop the
probability-based tools that are at the basis of statistical inference. However, probability can also play a critical role in decision making, a subject we explore in Chapter 22.
6 . 1 A S S I G N I N G P RO B A B I L I T Y
TO
EVENTS
To introduce probability, we must first define a random experiment.
Random Experiment
A random experiment is an action or process that leads to one of several
possible outcomes.
Here are six illustrations of random experiments and their outcomes.
Illustration 1. Experiment:
Outcomes:
Flip a coin.
Heads and tails
Illustration 2. Experiment:
Outcomes:
Record marks on a statistics test (out of 100).
Numbers between 0 and 100
Illustration 3. Experiment:
Outcomes:
Record grade on a statistics test.
A, B, C, D, and F
Illustration 4. Experiment:
Outcomes:
Record student evaluations of a course.
Poor, fair, good, very good, and excellent
Illustration 5. Experiment:
Outcomes:
Measure the time to assemble a computer.
Number whose smallest possible value is 0 seconds with no
predefined upper limit
Illustration 6. Experiment:
Record the party that a voter will vote for in an upcoming
election.
Party A, Party B, . . .
Outcomes:
The first step in assigning probabilities is to produce a list of the outcomes. The
listed outcomes must be exhaustive, which means that all possible outcomes must be
included. In addition, the outcomes must be mutually exclusive, which means that no
two outcomes can occur at the same time.
To illustrate the concept of exhaustive outcomes consider this list of the outcomes
of the toss of a die:
1
2
3
4
5
This list is not exhaustive, because we have omitted 6.
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The concept of mutual exclusiveness can be seen by listing the following outcomes
in illustration 2:
0–50
50–60
60–70
70–80
80–100
If these intervals include both the lower and upper limits, then these outcomes are not
mutually exclusive because two outcomes can occur for any student. For example, if a
student receives a mark of 70, both the third and fourth outcomes occur.
Note that we could produce more than one list of exhaustive and mutually exclusive outcomes. For example, here is another list of outcomes for illustration 3:
Pass and fail
A list of exhaustive and mutually exclusive outcomes is called a sample space and is
denoted by S. The outcomes are denoted by O1, O2, . . . , Ok.
Sample Space
A sample space of a random experiment is a list of all possible outcomes of
the experiment. The outcomes must be exhaustive and mutually exclusive.
Using set notation, we represent the sample space and its outcomes as
S = 5O1,O2, Á ,Ok6
Once a sample space has been prepared we begin the task of assigning probabilities
to the outcomes. There are three ways to assign probability to outcomes. However it is
done, there are two rules governing probabilities as stated in the next box.
Requirements of Probabilities
Given a sample space S {O1, O2, . . . , Ok}, the probabilities assigned to the
outcomes must satisfy two requirements.
1. The probability of any outcome must lie between 0 and 1; that is,
0 … P(Oi) … 1
for each i
[Note: P(Oi) is the notation we use to represent the probability of
outcome i.]
2. The sum of the probabilities of all the outcomes in a sample space must
be 1. That is,
k
a P(Oi) = 1
i=1
Three Approaches to Assigning Probabilities
The classical approach is used by mathematicians to help determine probability associated with games of chance. For example, the classical approach specifies that the
probabilities of heads and tails in the flip of a balanced coin are equal to each other.
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Because the sum of the probabilities must be 1, the probability of heads and the probability of tails are both 50%. Similarly, the six possible outcomes of the toss of a balanced
die have the same probability; each is assigned a probability of 1/6. In some experiments, it is necessary to develop mathematical ways to count the number of outcomes.
For example, to determine the probability of winning a lottery, we need to determine
the number of possible combinations. For details on how to count events, see Keller’s
website Appendix Counting Formulas.
The relative frequency approach defines probability as the long-run relative frequency with which an outcome occurs. For example, suppose that we know that of the
last 1,000 students who took the statistics course you’re now taking, 200 received a
grade of A. The relative frequency of A’s is then 200/1000 or 20%. This figure represents an estimate of the probability of obtaining a grade of A in the course. It is only an
estimate because the relative frequency approach defines probability as the “long-run”
relative frequency. One thousand students do not constitute the long run. The larger
the number of students whose grades we have observed, the better the estimate
becomes. In theory, we would have to observe an infinite number of grades to determine the exact probability.
When it is not reasonable to use the classical approach and there is no history of
the outcomes, we have no alternative but to employ the subjective approach. In the
subjective approach, we define probability as the degree of belief that we hold in the
occurrence of an event. An excellent example is derived from the field of investment. An
investor would like to know the probability that a particular stock will increase in value.
Using the subjective approach, the investor would analyze a number of factors associated with the stock and the stock market in general and, using his or her judgment,
assign a probability to the outcomes of interest.
Defining Events
An individual outcome of a sample space is called a simple event. All other events are
composed of the simple events in a sample space.
Event
An event is a collection or set of one or more simple events in a sample
space.
In illustration 2, we can define the event, achieve a grade of A, as the set of numbers
that lie between 80 and 100, inclusive. Using set notation, we have
A = {80, 81, 82, Á , 99, 100}
Similarly,
F = {0, 1, 2, Á , 48, 49}
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Probability of Events
We can now define the probability of any event.
Probability of an Event
The probability of an event is the sum of the probabilities of the simple
events that constitute the event.
For example, suppose that in illustration 3, we employed the relative frequency
approach to assign probabilities to the simple events as follows:
P(A) = .20
P(B) = .30
P(C) = .25
P(D) = .15
P(F) = .10
The probability of the event, pass the course, is
P(Pass the course) = P(A) + P(B) + P(C) + P(D) = .20 + .30 + .25 + .15 = .90
Interpreting Probability
No matter what method was used to assign probability, we interpret it using the relative
frequency approach for an infinite number of experiments. For example, an investor
may have used the subjective approach to determine that there is a 65% probability that
a particular stock’s price will increase over the next month. However, we interpret the
65% figure to mean that if we had an infinite number of stocks with exactly the same
economic and market characteristics as the one the investor will buy, 65% of them will
increase in price over the next month. Similarly, we can determine that the probability
of throwing a 5 with a balanced die is 1/6. We may have used the classical approach to
determine this probability. However, we interpret the number as the proportion of
times that a 5 is observed on a balanced die thrown an infinite number of times.
This relative frequency approach is useful to interpret probability statements such
as those heard from weather forecasters or scientists. You will also discover that this is
the way we link the population and the sample in statistical inference.
EXERCISES
6.1 The weather forecaster reports that the probability
6.3 A quiz contains a multiple-choice question with five
of rain tomorrow is 10%.
a. Which approach was used to arrive at this number?
b. How do you interpret the probability?
possible answers, only one of which is correct. A student plans to guess the answer because he knows
absolutely nothing about the subject.
a. Produce the sample space for each question.
b. Assign probabilities to the simple events in the
sample space you produced.
c. Which approach did you use to answer part (b)?
d. Interpret the probabilities you assigned in part (b).
6.2 A sportscaster states that he believes that the proba-
bility that the New York Yankees will win the World
Series this year is 25%.
a. Which method was used to assign that probability?
b. How would you interpret the probability?
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6.4 An investor tells you that in her estimation there is a
60% probability that the Dow Jones Industrial
Averages index will increase tomorrow.
a. Which approach was used to produce this figure?
b. Interpret the 60% probability.
6.5 The sample space of the toss of a fair die is
b. Use set notation to define the event, sell more
than three computers.
c. What is the probability of selling five computers?
d. What is the probability of selling two, three, or
four computers?
e. What is the probability of selling six computers?
6.9 Three contractors (call them contractors 1, 2, and 3)
S = {1, 2, 3, 4, 5, 6}
If the die is balanced each simple event has the same
probability. Find the probability of the following
events.
a. An even number
b. A number less than or equal to 4
c. A number greater than or equal to 5
6.6 Four candidates are running for mayor. The four
candidates are Adams, Brown, Collins, and Dalton.
Determine the sample space of the results of the
election.
6.7 Refer to Exercise 6.6. Employing the subjective
approach a political scientist has assigned the following probabilities:
P(Adams wins) .42
P(Brown wins) .09
P(Dalton wins) .22
Determine the probabilities of the following events.
a. Adams loses.
b. Either Brown or Dalton wins.
c. Adams, Brown, or Collins wins.
6.8 The manager of a computer store has kept track of
the number of computers sold per day. On the basis
of this information, the manager produced the following list of the number of daily sales.
0
1
2
3
4
5
Probability
.08
.17
.26
.21
.18
.10
a. If we define the experiment as observing the
number of computers sold tomorrow, determine
the sample space.
6 . 2 J O I N T , M A RG I N A L ,
6.10 Refer to Exercise 6.9. Suppose that you believe that
contractor 1 is twice as likely to win as contractor 3
and that contractor 2 is three times as likely to win as
contactor 3. What are the probabilities of winning
for each contractor?
6.11 Shoppers can pay for their purchases with cash, a
credit card, or a debit card. Suppose that the proprietor of a shop determines that 60% of her customers
use a credit card, 30% pay with cash, and the rest use
a debit card.
a. Determine the sample space for this experiment.
b. Assign probabilities to the simple events.
c. Which method did you use in part (b)?
6.12 Refer to Exercise 6.11.
P(Collins wins) .27
Number of Computers Sold
bid on a project to build a new bridge. What is the
sample space?
AND
a. What is the probability that a customer does not
use a credit card?
b. What is the probability that a customer pays in
cash or with a credit card?
6.13 A survey asks adults to report their marital status.
The sample space is S {single, married, divorced,
widowed}. Use set notation to represent the event
the adult is not married.
6.14 Refer to Exercise 6.13. Suppose that in the city in
which the survey is conducted, 50% of adults are
married, 15% are single, 25% are divorced, and
10% are widowed.
a. Assign probabilities to each simple event in the
sample space.
b. Which approach did you use in part (a)?
6.15 Refer to Exercises 6.13 and 6.14. Find the probabil-
ity of each of the following events.
a. The adult is single.
b. The adult is not divorced
c. The adult is either widowed or divorced.
C O N D I T I O N A L P RO B A B I L I T Y
In the previous section, we described how to produce a sample space and assign probabilities to the simple events in the sample space. Although this method of determining
probability is useful, we need to develop more sophisticated methods. In this section,
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we discuss how to calculate the probability of more complicated events from the probability of related events. Here is an illustration of the process.
The sample space for the toss of a die is
S {1, 2, 3, 4, 5, 6}
If the die is balanced, the probability of each simple event is 1/6. In most parlor
games and casinos, players toss two dice. To determine playing and wagering strategies,
players need to compute the probabilities of various totals of the two dice. For example,
the probability of tossing a total of 3 with two dice is 2/36. This probability was derived
by creating combinations of the simple events. There are several different types of combinations. One of the most important types is the intersection of two events.
Intersection
Intersection of Events A and B
The intersection of events A and B is the event that occurs when both A
and B occur. It is denoted as
A and B
The probability of the intersection is called the joint probability.
For example, one way to toss a 3 with two dice is to toss a 1 on the first die and a 2
on the second die, which is the intersection of two simple events. Incidentally, to compute the probability of a total of 3, we need to combine this intersection with another
intersection, namely, a 2 on the first die and a 1 on the second die. This type of combination is called a union of two events, and it will be described later in this section. Here
is another illustration.
APPLICATIONS in FINANCE
Mutual funds
© AP Photo/Charles Bennett
A mutual fund is a pool of investments made on behalf of people who share similar
objectives. In most cases, a professional manager who has been educated in
finance and statistics manages the fund. He or she makes decisions to buy and sell
individual stocks and bonds in accordance with a specified investment philosophy.
For example, there are funds that concentrate on other publicly traded mutual fund
companies. Other mutual funds specialize in Internet stocks (so-called dot-coms),
whereas others buy stocks of biotechnology firms. Surprisingly, most mutual funds do
not outperform the market; that is, the increase in the net asset value (NAV) of the mutual fund
is often less than the increase in the value of stock indexes that represent their stock markets. One
reason for this is the management expense ratio (MER) which is a measure of the costs charged to
the fund by the manager to cover expenses, including the salary and bonus of the managers. The
MERs for most funds range from .5% to more than 4%. The ultimate success of the fund depends
on the skill and knowledge of the fund manager. This raises the question, which managers do best?
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EXAMPLE
6.1
Determinants of Success among Mutual Fund
Managers—Part 1*
Why are some mutual fund managers more successful than others? One possible factor
is the university where the manager earned his or her master of business administration
(MBA). Suppose that a potential investor examined the relationship between how well
the mutual fund performs and where the fund manager earned his or her MBA. After
the analysis, Table 6.1, a table of joint probabilities, was developed. Analyze these probabilities and interpret the results.
TABLE
6.1
Determinants of Success among Mutual Fund Managers, Part 1*
MUTUAL FUND
OUTPERFORMS MARKET
MUTUAL FUND DOES
NOT OUTPERFORM MARKET
Top-20 MBA program
.11
.29
Not top-20 MBA program
.06
.54
Table 6.1 tells us that the joint probability that a mutual fund outperforms the market and that its manager graduated from a top-20 MBA program is .11; that is, 11% of
all mutual funds outperform the market and their managers graduated from a top-20
MBA program. The other three joint probabilities are defined similarly:
The probability that a mutual fund outperforms the market and its manager did not
graduate from a top-20 MBA program is .06.
The probability that a mutual fund does not outperform the market and its manager
graduated from a top-20 MBA program is .29.
The probability that a mutual fund does not outperform the market and its manager
did not graduate from a top-20 MBA program is .54.
To help make our task easier, we’ll use notation to represent the events. Let
A1 Fund manager graduated from a top-20 MBA program
A2 Fund manager did not graduate from a top-20 MBA program
B1 Fund outperforms the market
B2 Fund does not outperform the market
Thus,
P(A1 and B1) .11
P(A2 and B1) .06
P(A1 and B2) .29
P(A2 and B2) .54
*This example is adapted from “Are Some Mutual Fund Managers Better than Others? Cross-Sectional
Patterns in Behavior and Performance” by Judith Chevalier and Glenn Ellison, Working paper 5852,
National Bureau of Economic Research.
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Marginal Probability
The joint probabilities in Table 6.1 allow us to compute various probabilities. Marginal
probabilities, computed by adding across rows or down columns, are so named because
they are calculated in the margins of the table.
Adding across the first row produces
P(A1 and B1) + P(A1and B2) = .11 + .29 = .40
Notice that both intersections state that the manager graduated from a top-20 MBA program (represented by A1). Thus, when randomly selecting mutual funds, the probability
that its manager graduated from a top-20 MBA program is .40. Expressed as relative frequency, 40% of all mutual fund managers graduated from a top-20 MBA program.
Adding across the second row:
P(A2 and B1) + P(A2 and B2) = .06 + .54 = .60
This probability tells us that 60% of all mutual fund managers did not graduate from a
top-20 MBA program (represented by A2). Notice that the probability that a mutual
fund manager graduated from a top-20 MBA program and the probability that the
manager did not graduate from a top-20 MBA program add to 1.
Adding down the columns produces the following marginal probabilities.
Column 1:
P(A1 and B1) + P(A2 and B1) = .11 + .06 = .17
Column 2:
P(A1 and B2) + P(A2 and B2) = .29 + .54 = .83
These marginal probabilities tell us that 17% of all mutual funds outperform the market and that 83% of mutual funds do not outperform the market.
Table 6.2 lists all the joint and marginal probabilities.
TABLE
6. 2
Joint and Marginal Probabilities
MUTUAL FUND
OUTPERFORMS
MARKET
MUTUAL FUND DOES
NOT OUTPERFORM
MARKET
TOTALS
Top-20 MBA program
P(A1 and B1) .11
P(A1 and B2) .29
P(A1) .40
Not top-20 MBA
program
P(A2 and B1) .06
P(A2 and B2) .54
P(A2) .60
P(B1) .17
P(B2) .83
1.00
Totals
Conditional Probability
We frequently need to know how two events are related. In particular, we would like to
know the probability of one event given the occurrence of another related event. For
example, we would certainly like to know the probability that a fund managed by a
graduate of a top-20 MBA program will outperform the market. Such a probability will
allow us to make an informed decision about where to invest our money. This probability is called a conditional probability because we want to know the probability that a
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fund will outperform the market given the condition that the manager graduated from a
top-20 MBA program. The conditional probability that we seek is represented by
P(B1 ƒ A1)
where the “|” represents the word given. Here is how we compute this conditional
probability.
The marginal probability that a manager graduated from a top-20 MBA program is
.40, which is made up of two joint probabilities. They are (1) the probability that the
mutual fund outperforms the market and the manager graduated from a top-20 MBA
program [P(A1 and B1)] and (2) the probability that the fund does not outperform the
market and the manager graduated from a top-20 MBA program [P(A1 and B2)]. Their
joint probabilities are .11 and .29, respectively. We can interpret these numbers in the
following way. On average, for every 100 mutual funds, 40 will be managed by a graduate of a top-20 MBA program. Of these 40 managers, on average 11 of them will manage a mutual fund that will outperform the market. Thus, the conditional probability is
11/40 = .275. Notice that this ratio is the same as the ratio of the joint probability to the
marginal probability .11/.40. All conditional probabilities can be computed this way.
Conditional Probability
The probability of event A given event B is
P(A ƒ B) =
P(A and B)
P(B)
The probability of event B given event A is
P(B ƒ A) =
EXAMPLE
6. 2
P(A and B)
P(A)
Determinants of Success among Mutual Fund
Managers—Part 2
Suppose that in Example 6.1 we select one mutual fund at random and discover that it
did not outperform the market. What is the probability that a graduate of a top-20
MBA program manages it?
SOLUTION
We wish to find a conditional probability. The condition is that the fund did not outperform the market (event B2), and the event whose probability we seek is that the fund
is managed by a graduate of a top-20 MBA program (event A1). Thus, we want to compute the following probability:
P(A1 ƒ B2)
Using the conditional probability formula, we find
P(A1 ƒ B2) =
P(A1 and B2)
P(B2)
=
.29
= .349
.83
Thus, 34.9% of all mutual funds that do not outperform the market are managed by
top-20 MBA program graduates.
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The calculation of conditional probabilities raises the question of whether the two
events, the fund outperformed the market and the manager graduated from a top-20
MBA program, are related, a subject we tackle next.
Independence
One of the objectives of calculating conditional probability is to determine whether two
events are related. In particular, we would like to know whether they are independent
events.
Independent Events
Two events A and B are said to be independent if
P(A ƒ B) = P(A)
or
P(B ƒ A) = P(B)
Put another way, two events are independent if the probability of one event is not
affected by the occurrence of the other event.
EXAMPLE
6. 3
Determinants of Success among Mutual Fund
Managers—Part 3
Determine whether the event that the manager graduated from a top-20 MBA program
and the event the fund outperforms the market are independent events.
SOLUTION
We wish to determine whether A1 and B1 are independent. To do so, we must calculate
the probability of A1 given B1; that is,
P(A1 ƒ B1) =
P(A1 and B1)
P(B1)
=
.11
= .647
.17
The marginal probability that a manager graduated from a top-20 MBA program is
P(A1) = .40
Since the two probabilities are not equal, we conclude that the two events are dependent.
Incidentally, we could have made the decision by calculating P(B1 ƒ A1) = .275 and
observing that it is not equal to P(B1) .17.
Note that there are three other combinations of events in this problem. They are
(A1 and B2), (A2 and B1), (A2 and B2) [ignoring mutually exclusive combinations (A1 and
A2) and (B1 and B2), which are dependent]. In each combination, the two events are
dependent. In this type of problem, where there are only four combinations, if one
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combination is dependent, then all four will be dependent. Similarly, if one combination is independent, then all four will be independent. This rule does not apply to any
other situation.
Union
Another event that is the combination of other events is the union.
Union of Events A and B
The union of events A and B is the event that occurs when either A or B or
both occur. It is denoted as
A or B
EXAMPLE
6. 4
Determinants of Success among Mutual Fund
Managers—Part 4
Determine the probability that a randomly selected fund outperforms the market or the
manager graduated from a top-20 MBA program.
SOLUTION
We want to compute the probability of the union of two events
P(A1 or B1)
The union A1 or B1 consists of three events; That is, the union occurs whenever any of
the following joint events occurs:
1. Fund outperforms the market and the manager graduated from a top-20 MBA
program
2. Fund outperforms the market and the manager did not graduate from a top-20
MBA program
3. Fund does not outperform the market and the manager graduated from a top-20
MBA program
Their probabilities are
P(A1 and B1) .11
P(A2 and B1) .06
P(A1 and B2) .29
Thus, the probability of the union—the fund outperforms the market or the manager
graduated from a top-20 MBA program—is the sum of the three probabilities; That is,
P(A1 or B1) = P(A1 and B1) + P(A2 and B1) + P(A1 and B2) = .11 + .06 + .29 = .46
Notice that there is another way to produce this probability. Of the four probabilities in Table 6.1, the only one representing an event that is not part of the union is the
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probability of the event the fund does not outperform the market and the manager did
not graduate from a top-20 MBA program. That probability is
P(A2 and B2) = .54
which is the probability that the union does not occur. Thus, the probability of the union
is
P(A1 or B1) = 1 - P(A2 and B2) = 1 - .54 = .46.
Thus, we determined that 46% of mutual funds either outperform the market or are
managed by a top-20 MBA program graduate or have both characteristics.
EXERCISES
6.16 Given the following table of joint probabilities,
calculate the marginal probabilities.
B1
B2
A1
A2
A3
.1
.2
.3
.1
.2
.1
B1
B2
B1
B2
A1
A2
.4
.2
.3
.1
6.18 Refer to Exercise 6.17.
a. Determine P(A1 ƒ B1).
b. Determine P(A2 ƒ B1).
c. Did your answers to parts (a) and (b) sum to 1? Is
this a coincidence? Explain.
6.19 Refer to Exercise 6.17. Calculate the following
probabilities.
a. P(A1 ƒ B2)
b. P(B2 ƒ A1)
c. Did you expect the answers to parts (a) and (b) to
be reciprocals? In other words, did you expect
that P(A1 ƒ B2) = 1/ P(B2 ƒ A1)? Why is this impossible (unless both probabilities are 1)?
6.20 Are the events in Exercise 6.17 independent?
Explain.
6.21 Refer to Exercise 6.17. Compute the following.
a. P(A1 or B1)
b. P(A1 or B2)
c. P(A1 or A2)
.20
.05
.60
.15
from the following joint probabilities.
B1
B2
A1
A2
.20
.60
.15
.05
6.24 Suppose we have the following joint probabilities.
B1
B2
A1
A2
A3
.15
.25
.20
.25
.10
.05
Compute the marginal probabilities.
6.25 Refer to Exercise 6.24.
a. Compute P(A2 ƒ B2).
b. Compute P(B2 ƒ A2).
c. Compute P(B1 ƒ A2).
6.26 Refer to Exercise 6.24.
a. Compute P(A1 or A2).
b. Compute P(A2 or B2).
c. Compute P(A3 or B1).
6.27 Discrimination in the workplace is illegal, and com-
panies that discriminate are often sued. The female
instructors at a large university recently lodged a
complaint about the most recent round of promotions from assistant professor to associate professor.
An analysis of the relationship between gender and
promotion produced the following joint probabilities.
6.22 Suppose that you have been given the following
joint probabilities. Are the events independent?
Explain.
A2
6.23 Determine whether the events are independent
6.17 Calculate the marginal probabilities from the fol-
lowing table of joint probabilities.
A1
Female
Male
Promoted
Not Promoted
.03
.17
.12
.68
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a. What is the rate of promotion among female
assistant professors?
b. What is the rate of promotion among male assistant professors?
c. Is it reasonable to accuse the university of gender
bias?
One 60- to 65-year-old man is selected at random.
What is the probability of the following events?
a. He is a smoker.
b. He does not have lung disease.
c. He has lung disease given that he is a smoker.
d. He has lung disease given that he does not smoke.
6.28 A department store analyzed its most recent sales
6.31 Refer to Exercise 6.30. Are smoking and lung dis-
and determined the relationship between the way
the customer paid for the item and the price category of the item. The joint probabilities in the following table were calculated.
Cash
Less than $20
$20–$100
More than $100
Credit Card Debit Card
.09
.05
.03
.03
.21
.23
.04
.18
.14
a. What proportion of purchases was paid by debit
card?
b. Find the probability that a credit card purchase
was more than $100.
c. Determine the proportion of purchases made by
credit card or by debit card.
6.29 The following table lists the probabilities of unem-
ployed females and males and their educational
attainment.
Female Male
Less than high school
High school graduate
Some college or university—no degree
College or university graduate
.077
.154
.141
.092
.110
.201
.129
.096
(Source: Statistical Abstract of the United States, 2009, Table 607.)
a. If one unemployed person is selected at random,
what is the probability that he or she did not finish high school?
b. If an unemployed female is selected at random,
what is the probability that she has a college or
university degree?
c. If an unemployed high school graduate is
selected at random, what is the probability that
he is a male?
6.30 The costs of medical care in North America are
increasing faster than inflation, and with the baby
boom generation soon to need health care, it
becomes imperative that countries find ways to
reduce both costs and demand. The following table
lists the joint probabilities associated with smoking
and lung disease among 60- to 65-year-old men.
He has lung disease
He does not have lung disease
He is a
smoker
He is a
nonsmoker
.12
.19
.03
.66
ease among 60- to 65-year-old men related?
6.32 The method of instruction in college and university
applied statistics courses is changing. Historically,
most courses were taught with an emphasis on manual
calculation. The alternative is to employ a computer
and a software package to perform the calculations. An
analysis of applied statistics courses investigated
whether the instructor’s educational background is
primarily mathematics (or statistics) or some other
field. The result of this analysis is the accompanying
table of joint probabilities.
Statistics Course
Statistics Course
Emphasizes Manual Employs Computer
Calculations
and Software
Mathematics or
statistics education
Other education
.23
.11
.36
.30
a. What is the probability that a randomly selected
applied statistics course instructor whose education
was in statistics emphasizes manual calculations?
b. What proportion of applied statistics courses
employ a computer and software?
c. Are the educational background of the instructor and the way his or her course is taught
independent?
6.33 A restaurant chain routinely surveys its customers.
Among other questions, the survey asks each customer
whether he or she would return and to rate the quality
of food. Summarizing hundreds of thousands of questionnaires produced this table of joint probabilities.
Rating
Poor
Fair
Good
Excellent
Customer
Will Return
Customer Will
Not Return
.02
.08
.35
.20
.10
.09
.14
.02
a. What proportion of customers say that they will
return and rate the restaurant’s food as good?
b. What proportion of customers who say that they
will return rate the restaurant’s food as good?
c. What proportion of customers who rate the
restaurant’s food as good say that they will
return?
d. Discuss the differences in your answers to parts
(a), (b), and (c).
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PROBABILITY
6.34 To determine whether drinking alcoholic beverages
has an effect on the bacteria that cause ulcers,
researchers developed the following table of joint
probabilities.
Number of Alcoholic
Drinks per Day
None
One
Two
More than two
Ulcer
No Ulcer
.01
.03
.03
.04
.22
.19
.32
.16
a. What proportion of people have ulcers?
b. What is the probability that a teetotaler (no alcoholic beverages) develops an ulcer?
c. What is the probability that someone who has an
ulcer does not drink alcohol?
d. What is the probability that someone who has an
ulcer drinks alcohol?
189
Watch
Violent
Movie
Watch
Nonviolent
Movie
.15
.35
.18
.32
Remember the brand name
Do not remember the brand name
a. What proportion of viewers remember the brand
name?
b. What proportion of viewers who watch the violent movie remember the brand name?
c. Does watching a violent movie affect whether the
viewer will remember the brand name? Explain.
6.37 Is there a relationship between the male hormone
testosterone and criminal behavior? To answer this
question, medical researchers measured the testosterone level of penitentiary inmates and recorded
whether they were convicted of murder. After analyzing the results, the researchers produced the
following table of joint probabilities.
6.35 An analysis of fired or laid-off workers, their age,
and the reasons for their departure produced the following table of joint probabilities.
Age Category
Reason for job loss
Plant or company
closed or moved
Insufficient work
Position or shift
abolished
20–24
25–54
55–64
65 and
older
.015
.014
.320
.180
.089
.034
.029
.011
.006
.214
.071
.016
(Source: Statistical Abstract of the United States, 2009, Table 593.)
a. What is the probability that a 25- to 54-year-old
employee was laid off or fired because of insufficient work?
b. What proportion of laid-off or fired workers is
age 65 and older?
c. What is the probability that a laid-off or fired
worker because the plant or company closed is 65
or older?
6.36 Many critics of television claim that there is too
much violence and that it has a negative effect on
society. There may also be a negative effect on
advertisers. To examine this issue, researchers developed two versions of a cops-and-robbers madefor-television movie. One version depicted several
violent crimes, and the other removed these scenes.
In the middle of the movie, one 60-second commercial was shown advertising a new product and brand
name. At the end of the movie, viewers were asked
to name the brand. After observing the results, the
researchers produced the following table of joint
probabilities.
Testosterone Level
Murderer
Other Felon
.27
.21
.24
.28
Above average
Below average
a. What proportion of murderers have above-average
testosterone levels?
b. Are levels of testosterone and the crime committed independent? Explain.
6.38 The issue of health care coverage in the United
States is becoming a critical issue in American politics. A large-scale study was undertaken to determine who is and is not covered. From this study,
the following table of joint probabilities was
produced.
Age
Category
25–34
35–44
45–54
55–64
Has Health
Insurance
Does Not Have
Health Insurance
.167
.209
.225
.177
.085
.061
.049
.026
(Source: U.S. Department of Health and Human Services.)
If one person is selected at random, find the following probabilities.
a. P(Person has health insurance)
b. P(Person 55–64 has no health insurance)
c. P(Person without health insurance is between
25 and 34 years old)
6.39 Violent crime in many American schools is an
unfortunate fact of life. An analysis of schools and
violent crime yielded the table of joint probabilities
shown next.
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Level
Violent Crime
Committed
This Year
No Violent Crime
Committed
.393
.176
.134
.074
.191
.010
.007
.015
Primary
Middle
High School
Combined
Industry
(Source: Statistical Abstract of the United States, 2009, Table 237.)
Number of
Employees
Fewer than 20
20 to 99
100 or more
Construction
Manufacturing
Retail
.464
.039
.005
.147
.049
.019
.237
.035
.005
(Source: Statistical Abstract of the United States, 2009, Table 737.)
If one school is randomly selected find the following
probabilities.
a. Probability of at least one incident of violent
crime during the year in a primary school
b. Probability of no violent crime during the year
If one firm is selected at random, find the probability of the following events.
a. The firm employs fewer than 20 employees.
b. The firm is in the retail industry.
c. A firm in the construction industry employs
between 20 and 99 workers.
6.40 Refer to Exercise 6.39. A similar analysis produced
6.43 Credit scorecards are used by financial institutions
these joint probabilities.
Enrollment
Violent Crime
Committed
This Year
No Violent Crime
Committed
.159
.221
.289
.108
.091
.065
.063
.004
Less than 300
300 to 499
500 to 999
1,000 or more
(Source: Statistical Abstract of the United States, 2009, Table 237.)
a. What is the probability that a school with an
enrollment of less than 300 had at least one violent crime during the year?
b. What is the probability that a school that has at
least one violent crime had an enrollment of less
than 300?
6.41 A firm has classified its customers in two ways:
(1) according to whether the account is overdue
and (2) whether the account is new (less than
12 months) or old. An analysis of the firm’s records
provided the input for the following table of joint
probabilities.
New
Old
Overdue
Not Overdue
.06
.52
.13
.29
One account is randomly selected.
a. If the account is overdue, what is the probability
that it is new?
b. If the account is new, what is the probability that
it is overdue?
c. Is the age of the account related to whether it is
overdue? Explain.
6.42 How are the size of a firm (measured in terms of the
number of employees) and the type of firm related?
To help answer the question, an analyst referred to
the U.S. Census and developed the following table
of joint probabilities.
to help decide to whom loans should be granted (see
the Applications in Banking: Credit Scorecards summary on page 63). An analysis of the records of one
bank produced the following probabilities.
Score
Loan Performance
Under 400
400 or More
.19
.13
.64
.04
Fully repaid
Defaulted
a. What proportion of loans are fully repaid?
b. What proportion of loans given to scorers of less
than 400 fully repay?
c. What proportion of loans given to scorers of 400
or more fully repay?
d. Are score and whether the loan is fully repaid
independent? Explain.
6.44 A retail outlet wanted to know whether its weekly
advertisement in the daily newspaper works. To
acquire this critical information, the store manager
surveyed the people who entered the store and
determined whether each individual saw the ad and
whether a purchase was made. From the information developed, the manager produced the following
table of joint probabilities. Are the ads effective?
Explain.
See ad
Do not see ad
Purchase
No Purchase
.18
.12
.42
.28
6.45 To gauge the relationship between education and
unemployment, an economist turned to the U.S.
Census from which the following table of joint
probabilities was produced.
Education
Not a high school graduate
High school graduate
Employed
Unemployed
.091
.282
.008
.014
(Continued)
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PROBABILITY
Some college, no degree
Associate’s degree
Bachelor’s degree
Advanced degree
.166
.095
.213
.115
.007
.003
.004
.002
(Source: Statistical Abstract of the United States, 2009, Table 223.)
a. What is the probability that a high school graduate is unemployed?
b. Determine the probability that a randomly
selected individual is employed.
c. Find the probability that an unemployed person
possesses an advanced degree.
d. What is the probability that a randomly selected
person did not finish high school?
Region
Education
Not a high
school graduate
High school
graduate
Some college,
no degree
Associate’s degree
Bachelor’s degree
Advanced degree
a major one for most companies. One factor that
is often considered is the education level of the
location’s residents. Census information may
be useful in this regard. After analyzing a recent
census, a company produced the following joint
probabilities.
AND
Northeast
Midwest
South
West
.024
.024
.059
.036
.063
.078
.117
.059
.023
.015
.038
.024
.039
.021
.040
.020
.061
.030
.065
.032
.045
.020
.046
.023
(Source: Statistical Abstract of the United States, 2009, Table 223.)
6.46 The decision about where to build a new plant is
6 . 3 P RO B A B I L I T Y R U L E S
191
a. Determine the probability that a person living in
the West has a bachelor’s degree.
b. Find the probability that a high school graduate
lives in the Northeast.
c. What is the probability that a person selected at
random lives in the South?
d. What is the probability that a person selected at
random does not live in the South?
TREES
In Section 6.2, we introduced intersection and union and described how to determine the
probability of the intersection and the union of two events. In this section, we present other
methods of determining these probabilities. We introduce three rules that enable us to calculate the probability of more complex events from the probability of simpler events.
Complement Rule
The complement of event A is the event that occurs when event A does not occur. The
complement of event A is denoted by AC. The complement rule defined here derives
from the fact that the probability of an event and the probability of the event’s complement must sum to 1.
Complement Rule
P(AC ) = 1 - P(A)
for any event A.
We will demonstrate the use of this rule after we introduce the next rule.
Multiplication Rule
The multiplication rule is used to calculate the joint probability of two events. It is
based on the formula for conditional probability supplied in the previous section; that
is, from the following formula
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P(A ƒ B) =
P(A and B)
P(B)
we derive the multiplication rule simply by multiplying both sides by P(B).
Multiplication Rule
The joint probability of any two events A and B is
P(A and B) = P(B)P(A ƒ B)
or, altering the notation,
P(A and B) = P(A)P(B ƒ A)
If A and B are independent events, P(A ƒ B) = P(A) and P(B ƒ A) = P(B). It follows that
the joint probability of two independent events is simply the product of the probabilities of the two events. We can express this as a special form of the multiplication rule.
Multiplication Rule for Independent Events
The joint probability of any two independent events A and B is
P(A and B) = P(A)P(B)
EXAMPLE
6. 5 *
Selecting Two Students without Replacement
A graduate statistics course has seven male and three female students. The professor
wants to select two students at random to help her conduct a research project. What is
the probability that the two students chosen are female?
SOLUTION
Let A represent the event that the first student chosen is female and B represent the
event that the second student chosen is also female. We want the joint probability
P(A and B). Consequently, we apply the multiplication rule:
P(A and B) = P(A)P(B ƒ A)
Because there are 3 female students in a class of 10, the probability that the first student
chosen is female is
P(A) = 3/10
*This example can be solved using the Hypergeometric distribution, which is described in the Keller’s
website Appendix Hypergeometric Distribution.
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PROBABILITY
193
After the first student is chosen, there are only nine students left. Given that the first
student chosen was female, there are only two female students left. It follows that
P(B ƒ A) = 2/9
Thus, the joint probability is
P(A and B) = P(A)P(B ƒ A) = a
EXAMPLE
6.6
2
6
3
ba b =
= .067
10 9
90
Selecting Two Students with Replacement
Refer to Example 6.5. The professor who teaches the course is suffering from the flu
and will be unavailable for two classes. The professor’s replacement will teach the next
two classes. His style is to select one student at random and pick on him or her to
answer questions during that class. What is the probability that the two students chosen
are female?
SOLUTION
The form of the question is the same as in Example 6.5: We wish to compute the probability of choosing two female students. However, the experiment is slightly different. It
is now possible to choose the same student in each of the two classes taught by the
replacement. Thus, A and B are independent events, and we apply the multiplication
rule for independent events:
P(A and B) = P(A)P(B)
The probability of choosing a female student in each of the two classes is the same; that
is,
P(A) = 3/10 and P(B) = 3/10
Hence,
P(A and B) = P(A)P(B) = a
3
3
9
ba b =
= .09
10 10
100
Addition Rule
The addition rule enables us to calculate the probability of the union of two events.
Addition Rule
The probability that event A, or event B, or both occur is
P(A or B) = P(A) + P(B) - P(A and B)
If you’re like most students, you’re wondering why we subtract the joint probability
from the sum of the probabilities of A and B. To understand why this is necessary, examine Table 6.2 (page 183 ), which we have reproduced here as Table 6.3.
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TABLE
6. 3
Joint and Marginal Probabilities
B1
B2
TOTALS
A1
P(A1 and B1) .11
P(A1 and B2) .29
P(A1) .40
A2
P(A2 and B1) .06
P(A2 and B2) .54
P(A2) .60
P(B1) .17
P(B2) .83
1.00
Totals
This table summarizes how the marginal probabilities were computed. For example,
the marginal probability of A1 and the marginal probability of B1 were calculated as
P(A1) = P(A1 and B1) + P(A1 and B2) = .11 + .29 = .40
P(B1) = P(A1 and B1) + P(A2 and B1) = .11 + .06 = .17
If we now attempt to calculate the probability of the union of A1 and B1 by summing
their probabilities, we find
P(A1) + P(B1) = .11 + .29 + .11 + .06
Notice that we added the joint probability of A1 and B1 (which is .11) twice. To correct
the double counting, we subtract the joint probability from the sum of the probabilities
of A1and B1. Thus,
P(A1 or B1) = P(A1) + P(B1) - P(A1 and B1)
= [.11 + .29] + [.11 + .06] - .11
= .40 + .17 - .11 = .46
This is the probability of the union of A1 and B1, which we calculated in Example 6.4
(page 186).
As was the case with the multiplication rule, there is a special form of the addition
rule. When two events are mutually exclusive (which means that the two events cannot
occur together), their joint probability is 0.
Addition Rule for Mutually Exclusive Events
The probability of the union of two mutually exclusive events A and B is
P(A or B) = P(A) + P(B)
EXAMPLE
6.7
Applying the Addition Rule
In a large city, two newspapers are published, the Sun and the Post. The circulation
departments report that 22% of the city’s households have a subscription to the Sun
and 35% subscribe to the Post. A survey reveals that 6% of all households subscribe
to both newspapers. What proportion of the city’s households subscribe to either
newspaper?
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195
SOLUTION
We can express this question as, what is the probability of selecting a household at random that subscribes to the Sun, the Post, or both? Another way of asking the question is,
what is the probability that a randomly selected household subscribes to at least one of
the newspapers? It is now clear that we seek the probability of the union, and we must
apply the addition rule. Let A the household subscribes to the Sun and B the
household subscribes to the Post. We perform the following calculation:
P(A or B) = P(A) + P(B) - P(A and B) = .22 + .35 - .06 = .51
The probability that a randomly selected household subscribes to either newspaper is
.51. Expressed as relative frequency, 51% of the city’s households subscribe to either
newspaper.
Probability Trees
An effective and simpler method of applying the probability rules is the probability tree,
wherein the events in an experiment are represented by lines. The resulting figure
resembles a tree, hence the name. We will illustrate the probability tree with several
examples, including two that we addressed using the probability rules alone.
In Example 6.5, we wanted to find the probability of choosing two female students, where the two choices had to be different. The tree diagram in Figure 6.1
describes this experiment. Notice that the first two branches represent the two possibilities, female and male students, on the first choice. The second set of branches represents the two possibilities on the second choice. The probabilities of female and
male student chosen first are 3/10 and 7/10, respectively. The probabilities for the
second set of branches are conditional probabilities based on the choice of the first
student selected.
We calculate the joint probabilities by multiplying the probabilities on the
linked branches. Thus, the probability of choosing two female students is P(F and F)
(3/10)(2/9) 6/90. The remaining joint probabilities are computed similarly.
FIGURE
6.1
Probability Tree for Example 6.5
First choice
Second choice
2
F |F –
9
3
F ––
10
M| F 7–
9
3
F |M –
7
M ––
9
10
6
M| M –
9
Joint probability
( ) ( 29– ) = 90––6
7
3
21
–– ) ( – ) = ––
F and M: ( 10
9
90
F and F:
3
––
10
( ) ( 39– ) = 2190––
6
7
42
–– ) ( – ) = ––
M and M: ( 10
9
90
7
––
M and F: 10
In Example 6.6, the experiment was similar to that of Example 6.5. However, the
student selected on the first choice was returned to the pool of students and was eligible
to be chosen again. Thus, the probabilities on the second set of branches remain the
same as the probabilities on the first set, and the probability tree is drawn with these
changes, as shown in Figure 6.2.
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CHAPTER 6
FIGURE
6.2
Probability Tree for Example 6.6
First choice
Second choice
3
F ––
10
3
F ––
10
7
M ––
10
3
F ––
10
7
M ––
10
7
M ––
10
Joint probability
3
––
10
9
–––
( ) ( ) = 100
3
7
21
–– ) ( –– ) = –––
F and M: ( 10
10
100
F and F:
3
––
10
7
––
M and F: 10
21
–––
( ) ( 10––3 ) = 100
7
49
7
–– ) ( –– ) = –––
M and M: ( 10
10
100
The advantage of a probability tree on this type of problem is that it restrains its
users from making the wrong calculation. Once the tree is drawn and the probabilities
of the branches inserted, virtually the only allowable calculation is the multiplication of
the probabilities of linked branches. An easy check on those calculations is available.
The joint probabilities at the ends of the branches must sum to 1 because all possible
events are listed. In both figures, notice that the joint probabilities do indeed sum to 1.
The special form of the addition rule for mutually exclusive events can be applied
to the joint probabilities. In both probability trees, we can compute the probability that
one student chosen is female and one is male simply by adding the joint probabilities.
For the tree in Example 6.5, we have
P(F and M) + P(M and F) = 21/90 + 21/90 = 42/90
In the probability tree in Example 6.6, we find
P(F and M) + P(M and F) = 21/100 + 21/100 = 42/100
EXAMPLE
6.8
Probability of Passing the Bar Exam
Students who graduate from law schools must still pass a bar exam before becoming
lawyers. Suppose that in a particular jurisdiction the pass rate for first-time test takers
is 72%. Candidates who fail the first exam may take it again several months later. Of
those who fail their first test, 88% pass their second attempt. Find the probability that a
randomly selected law school graduate becomes a lawyer. Assume that candidates cannot take the exam more than twice.
SOLUTION
The probability tree in Figure 6.3 is employed to describe the experiment. Note that we
use the complement rule to determine the probability of failing each exam.
FIGURE
6.3
Probability Tree for Example 6.8
First exam
Second exam
Joint probability
Pass
Pass .72
.72
Pass|Fail .88
Fail and Pass (.28)(.88) = .2464
Fail|Fail .12
Fail and Fail (.28)(.12) = .0336
Fail .28
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We apply the multiplication rule to calculate P(Fail and Pass), which we find to be
.2464. We then apply the addition rule for mutually exclusive events to find the probability of passing the first or second exam:
P(Pass [on first exam]) + P(Fail [on first exam] and Pass [on second exam])
= .72 + .2464 = .9664
Thus, 96.64% of applicants become lawyers by passing the first or second exam.
EXERCISES
6.47 Given the following probabilities, compute all joint
probabilities.
P(A) = .9
P(AC ) = .1
P(B ƒ A) = .4
P(B ƒ AC ) = .7
6.48 Determine all joint probabilities from the following.
P(A) = .8
P(AC ) = .2
P(B ƒ A) = .4
P(B ƒ AC ) = .7
6.49 Draw a probability tree to compute the joint proba-
bilities from the following probabilities.
P(A) = .5
P(AC ) = .2
P(B ƒ A) = .4
P(B ƒ AC ) = .7
6.50 Given the following probabilities, draw a probability
tree to compute the joint probabilities.
P(A) = .8
P(AC ) = .2
P(B ƒ A) = .3
P(B ƒ AC ) = .3
6.51 Given the following probabilities, find the joint
probability P(A and B).
P(A) = .7
P(B ƒ A) = .3
6.52 Approximately 10% of people are left-handed. If
two people are selected at random, what is the probability of the following events?
a. Both are right-handed.
b. Both are left-handed.
c. One is right-handed and the other is left-handed.
d. At least one is right-handed.
6.53 Refer to Exercise 6.52. Suppose that three people
are selected at random.
a. Draw a probability tree to depict the experiment.
b. If we use the notation RRR to describe the selection of three right-handed people, what are the
descriptions of the remaining seven events? (Use
L for left-hander.)
c. How many of the events yield no right-handers,
one right-hander, two right-handers, three righthanders?
d. Find the probability of no right-handers, one
right-hander, two right-handers, three righthanders.
6.54 Suppose there are 100 students in your accounting
class, 10 of whom are left-handed. Two students are
selected at random.
a. Draw a probability tree and insert the probabilities for each branch.
What is the probability of the following events?
b. Both are right-handed.
c. Both are left-handed.
d. One is right-handed and the other is left-handed.
e. At least one is right-handed
6.55 Refer to Exercise 6.54. Suppose that three people
are selected at random.
a. Draw a probability tree and insert the probabilities of each branch.
b. What is the probability of no right-handers, one
right-hander, two right-handers, three righthanders?
6.56 An aerospace company has submitted bids on two
separate federal government defense contracts. The
company president believes that there is a 40%
probability of winning the first contract. If they win
the first contract, the probability of winning the second is 70%. However, if they lose the first contract,
the president thinks that the probability of winning
the second contract decreases to 50%.
a. What is the probability that they win both contracts?
b. What is the probability that they lose both contracts?
c. What is the probability that they win only one
contract?
6.57 A telemarketer calls people and tries to sell them a
subscription to a daily newspaper. On 20% of her
calls, there is no answer or the line is busy. She sells
subscriptions to 5% of the remaining calls. For what
proportion of calls does she make a sale?
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6.58 A foreman for an injection-molding firm admits that
on 10% of his shifts, he forgets to shut off the injection machine on his line. This causes the machine to
overheat, increasing the probability from 2% to
20% that a defective molding will be produced during the early morning run. What proportion of
moldings from the early morning run is defective?
6.59 A study undertaken by the Miami-Dade Supervisor
of Elections in 2002 revealed that 44% of registered
voters are Democrats, 37% are Republicans, and
19% are others. If two registered voters are selected
at random, what is the probability that both of them
have the same party affiliation? (Source: Miami
Herald, April 11, 2002.)
6.60 In early 2001, the U.S. Census Bureau started releas-
ing the results of the 2000 census. Among many
other pieces of information, the bureau recorded the
race or ethnicity of the residents of every county in
every state. From these results, the bureau calculated
a “diversity index” that measures the probability that
two people chosen at random are of different races or
ethnicities. Suppose that the census determined that
in a county in Wisconsin 80% of its residents are
white, 15% are black, and 5% are Asian. Calculate
the diversity index for this county.
6.61 A survey of middle-aged men reveals that 28% of
them are balding at the crown of their heads.
Moreover, it is known that such men have an 18%
probability of suffering a heart attack in the next
10 years. Men who are not balding in this way have
an 11% probability of a heart attack. Find the probability that a middle-aged man will suffer a heart
attack sometime in the next 10 years.
6.62 The chartered financial analyst (CFA) is a designa-
tion earned after a candidate has taken three annual
exams (CFA I, II, and III). The exams are taken in
early June. Candidates who pass an exam are eligible
to take the exam for the next level in the following
year. The pass rates for levels I, II, and III are
.57, .73, and .85, respectively. Suppose that 3,000
candidates take the level I exam, 2,500 take the level
II exam, and 2,000 take the level III exam. Suppose
that one student is selected at random. What is the
probability that he or she has passed the exam?
(Source: Institute of Financial Analysts.)
6.63 The Nickels restaurant chain regularly conducts sur-
veys of its customers. Respondents are asked to assess
food quality, service, and price. The responses are
Excellent
Good
Fair
Surveyed customers are also asked whether they
would come back. After analyzing the responses, an
expert in probability determined that 87% of customers say that they will return. Of those who so
indicate, 57% rate the restaurant as excellent, 36%
rate it as good, and the remainder rate it as fair. Of
those who say that they won’t return, the probabilities are 14%, 32%, and 54%, respectively. What
proportion of customers rate the restaurant as good?
6.64 Researchers at the University of Pennsylvania
School of Medicine have determined that children
under 2 years old who sleep with the lights on have a
36% chance of becoming myopic before they are 16.
Children who sleep in darkness have a 21% probability of becoming myopic. A survey indicates that
28% of children under 2 sleep with some light on.
Find the probability that a child under 16 is myopic.
6.65 All printed circuit boards (PCBs) that are manufac-
tured at a certain plant are inspected. An analysis of
the company’s records indicates that 22% of all PCBs
are flawed in some way. Of those that are flawed, 84%
are reparable and the rest must be discarded. If a
newly produced PCB is randomly selected, what is
the probability that it does not have to be discarded?
6.66 A financial analyst has determined that there is a
22% probability that a mutual fund will outperform
the market over a 1-year period provided that it outperformed the market the previous year. If only 15%
of mutual funds outperform the market during any
year, what is the probability that a mutual fund will
outperform the market 2 years in a row?
6.67 An investor believes that on a day when the Dow
Jones Industrial Average (DJIA) increases, the probability that the NASDAQ also increases is 77%. If
the investor believes that there is a 60% probability
that the DJIA will increase tomorrow, what is the
probability that the NASDAQ will increase as well?
6.68 The controls of an airplane have several backup sys-
tems or redundancies so that if one fails the plane
will continue to operate. Suppose that the mechanism that controls the flaps has two backups. If the
probability that the main control fails is .0001 and
the probability that each backup will fail is .01, what
is the probability that all three fail to operate?
6.69 According to TNS Intersearch, 69% of wireless web
users use it primarily for receiving and sending
e-mail. Suppose that three wireless web users are
selected at random. What is the probability that all
of them use it primarily for e-mail?
6.70 A financial analyst estimates that the probability that
the economy will experience a recession in the next
12 months is 25%. She also believes that if the economy encounters a recession, the probability that her
mutual fund will increase in value is 20%. If there is
no recession, the probability that the mutual fund
will increase in value is 75%. Find the probability
that the mutual fund’s value will increase.
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6 . 4 B AY E S ’ S L AW
Conditional probability is often used to gauge the relationship between two events. In
many of the examples and exercises you’ve already encountered, conditional probability
measures the probability that an event occurs given that a possible cause of the event
has occurred. In Example 6.2, we calculated the probability that a mutual fund outperforms the market (the effect) given that the fund manager graduated from a top-20
MBA program (the possible cause). There are situations, however, where we witness a
particular event and we need to compute the probability of one of its possible causes.
Bayes’s Law is the technique we use.
EXAMPLE
6 .9
Should an MBA Applicant Take a Preparatory Course?
The Graduate Management Admission Test (GMAT) is a requirement for all applicants of
MBA programs. A variety of preparatory courses are designed to help applicants improve
their GMAT scores, which range from 200 to 800. Suppose that a survey of MBA students
reveals that among GMAT scorers above 650, 52% took a preparatory course; whereas
among GMAT scorers of less than 650 only 23% took a preparatory course. An applicant
to an MBA program has determined that he needs a score of more than 650 to get into a
certain MBA program, but he feels that his probability of getting that high a score is quite
low—10%. He is considering taking a preparatory course that costs $500. He is willing to
do so only if his probability of achieving 650 or more doubles. What should he do?
SOLUTION
The easiest way to address this problem is to draw a tree diagram. The following notation will be used:
A
GMAT score is 650 or more
AC
GMAT score less than 650
B
Took preparatory course
BC Did not take preparatory course
The probability of scoring 650 or more is
P(A) = .10
The complement rule gives us
P(AC ) = 1 - .10 = .90
Conditional probabilities are
P(B ƒ A) = .52
and
P(B ƒ AC ) = .23
Again using the complement rule, we find the following conditional probabilities:
P(BC ƒ A) = 1 - .52 = .48
and
P(BC ƒ AC ) = 1 - .23 = .77
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We would like to determine the probability that he would achieve a GMAT score of 650
or more given that he took the preparatory course; that is, we need to compute
P(A ƒ B)
Using the definition of conditional probability (page 184), we have
P(A ƒ B) =
P(A and B)
P(B)
Neither the numerator nor the denominator is known. The probability tree (Figure 6.4)
will provide us with the probabilities.
FIGURE
6.4
Probability Tree for Example 6.9
GMAT
Preparatory course
B| A .52
A .10
B c | A .48
B| A c .23
c
A .90
B c | A c .77
Joint probability
A and B: (.10)(.52) = .052
c
A and B : (.10)(.48) = .048
c
A and B: (.90)(.23) = .207
Ac and B c: (.90)(.77) = .693
As you can see,
P(A and B) = (.10)(.52) = .052
P(AC and B) = (.90)(.23) = .207
and
P(B) = P(A and B) + P(AC and B) = .052 + .207 = .259
Thus,
P(A ƒ B) =
P(A and B)
P(B)
=
.052
= .201
.259
The probability of scoring 650 or more on the GMAT doubles when the preparatory
course is taken.
Thomas Bayes first employed the calculation of conditional probability as shown in
Example 6.9 during the 18th century. Accordingly, it is called Bayes’s Law.
The probabilities P(A) and P(AC) are called prior probabilities because they are
determined prior to the decision about taking the preparatory course. The conditional
probabilities are called likelihood probabilities for reasons that are beyond the mathematics in this book. Finally, the conditional probability P(A ƒ B) and similar conditional
probabilities P(AC ƒ B), P(A ƒ BC), and P(AC ƒ BC) are called posterior probabilities or
revised probabilities because the prior probabilities are revised after the decision
about taking the preparatory course.
You may be wondering why we did not get P(A ƒ B) directly. In other words, why not
survey people who took the preparatory course and ask whether they received a score of
650 or more? The answer is that using the likelihood probabilities and using Bayes’s Law
allows individuals to set their own prior probabilities, which can then be revised. For
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PROBABILITY
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example, another MBA applicant may assess her probability of scoring 650 or more as
.40. Inputting the new prior probabilities produces the following probabilities:
P(A and B) = (.40)(.52) = .208
P(AC and B) = (.60)(.23) = .138
P(B) = P(A and B) + P(ACand B) = .208 + .138 = .346
P(A and B)
.208
=
= .601
P(A ƒ B) =
P(B)
.346
The probability of achieving a GMAT score of 650 or more increases by a more
modest 50% (from .40 to .601).
Bayes’s Law Formula (Optional)
Bayes’s Law can be expressed as a formula for those who prefer an algebraic approach
rather than a probability tree. We use the following notation.
The event B is the given event and the events
A1, A2, Á , Ak
are the events for which prior probabilities are known; that is,
P(A1), P(A2), Á , P(Ak)
are the prior probabilities.
The likelihood probabilities are
P(B ƒ A1), P(B ƒ A2), Á , P(B ƒ Ak)
and
P(A1 ƒ B), P(A2 ƒ B), Á , P(Ak ƒ B)
are the posterior probabilities, which represent the probabilities we seek.
Bayes’s Law Formula
P(Ai ƒ B) =
P(Ai)P(B ƒ Ai)
P(A1)P(B ƒ A1) + P(A2)P(B ƒ A2) + Á + P(Ak)P(B ƒ Ak)
To illustrate the use of the formula, we’ll redo Example 6.9. We begin by defining
the events.
A1 = GMAT score is 650 or more
A2 = GMAT score less than 650
B = Take preparatory course
The probabilities are
P(A1) = .10
The complement rule gives us
P(A2) = 1 - .10 = .90
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Conditional probabilities are
P(B ƒ A1) = .52
and
P(B ƒ A2) = .23
Substituting the prior and likelihood probabilities into the Bayes’s Law formula
yields the following:
P(A1)P(B ƒ A1)
P(A1 ƒ B) =
=
P(A1)P(B ƒ A1) + P(A2)P(B ƒ A2)
(.10)(.52)
=
(.10)(.52) + (.90)(.23)
.052
.052
=
= .201
.052 + .207
.259
Auditing Tax Returns: Solution
We need to revise the prior probability that this return contains significant fraud. The tree shown in
Figure 6.5 details the calculation.
F = Tax return is fraudulent
F C = Tax return is honest
E0 = Tax return contains no expense deductions
© Gary Buss/Taxi/Getty Images
As you can see, the calculation of the Bayes’s Law formula produces the same
results as the probability tree.
E1 = Tax return contains one expense deduction
E2 = tax return contains two expense deductions
P (E1)
= P (F and E1) + P (F C and E1) = .0140 + .1710 = .1850
P (F ƒ E1) = P (F and E1)>P (E1) = .0140/ .1850 = .0757
The probability that this return is fraudulent is .0757.
FIGURE
6.5
Tax return
Probability Tree for Auditing Tax Returns
Expense deduction
E 0 | F .27
E 1 |F .28
F .05
E 2 |F .45
E 0| F
F
c
.95
c
.71
E 1 | F c .18
E 2| F
c
.11
Joint probability
F and E0: (.05)(.27) = .0135
F and E1: (.05)(.28) = .0140
F and E2: (.05)(.45) = .0225
c
F and E0: (.95)(.71) = .6745
c
F and E1: (.95)(.18) = .1710
F c and E2: (.95)(.11) = .1045
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Applications in Medicine and Medical Insurance (Optional)
Physicians routinely perform medical tests, called screenings, on their patients.
Screening tests are conducted for all patients in a particular age and gender group,
regardless of their symptoms. For example, men in their 50s are advised to take a
prostate-specific antigen (PSA) test to determine whether there is evidence of prostate
cancer. Women undergo a Pap test for cervical cancer. Unfortunately, few of these tests
are 100% accurate. Most can produce false-positive and false-negative results. A falsepositive result is one in which the patient does not have the disease, but the test shows
positive. A false-negative result is one in which the patient does have the disease, but
the test produces a negative result. The consequences of each test are serious and costly.
A false-negative test results in not detecting a disease in a patient, therefore postponing
treatment, perhaps indefinitely. A false-positive test leads to apprehension and fear for
the patient. In most cases, the patient is required to undergo further testing such as a
biopsy. The unnecessary follow-up procedure can pose medical risks.
False-positive test results have financial repercussions. The cost of the follow-up
procedure, for example, is usually far more expensive than the screening test. Medical
insurance companies as well as government-funded plans are all adversely affected by
false-positive test results. Compounding the problem is that physicians and patients are
incapable of properly interpreting the results. A correct analysis can save both lives and
money.
Bayes’s Law is the vehicle we use to determine the true probabilities associated with
screening tests. Applying the complement rule to the false-positive and false-negative
rates produces the conditional probabilities that represent correct conclusions. Prior
probabilities are usually derived by looking at the overall proportion of people with the
diseases. In some cases, the prior probabilities may themselves have been revised
because of heredity or demographic variables such as age or race. Bayes’s Law allows us
to revise the prior probability after the test result is positive or negative.
Example 6.10 is based on the actual false-positive and false-negative rates. Note
however, that different sources provide somewhat different probabilities. The differences may be the result of the way positive and negative results are defined or the way
technicians conduct the tests. Students who are affected by the diseases described in the
example and exercises should seek clarification from their physicians.
EXAMPLE
6.10
Probability of Prostate Cancer
Prostate cancer is the most common form of cancer found in men. The probability of
developing prostate cancer over a lifetime is 16%. (This figure may be higher since many
prostate cancers go undetected.) Many physicians routinely perform a PSA test, particularly for men over age 50. PSA is a protein produced only by the prostate gland and thus
is fairly easy to detect. Normally, men have PSA levels between 0 and 4 mg/ml. Readings
above 4 may be considered high and potentially indicative of cancer. However, PSA levels tend to rise with age even among men who are cancer free. Studies have shown that
the test is not very accurate. In fact, the probability of having an elevated PSA level given
that the man does not have cancer (false positive) is .135. If the man does have cancer, the
probability of a normal PSA level (false negative) is almost .300. (This figure may vary by
age and by the definition of high PSA level.) If a physician concludes that the PSA is high,
a biopsy is performed. Besides the concerns and health needs of the men, there are also
financial costs. The cost of the blood test is low (approximately $50). However, the cost
of the biopsy is considerably higher (approximately $1,000). A false-positive PSA test
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will lead to an unnecessary biopsy. Because the PSA test is so inaccurate, some private
and public medical plans do not pay for it. Suppose you are a manager in a medical insurance company and must decide on guidelines for whom should be routinely screened for
prostate cancer. An analysis of prostate cancer incidence and age produces the following
table of probabilities. (The probability of a man under 40 developing prostate cancer is
less than .0001, or small enough to treat as 0.)
Age
Probability of Developing
Prostate Cancer
40–49
50–59
60–69
70 and older
.010
.022
.046
.079
Assume that a man in each of the age categories undergoes a PSA test with a positive result. Calculate the probability that each man actually has prostate cancer and the
probability that he does not. Perform a cost–benefit analysis to determine the cost per
cancer detected.
SOLUTION
As we did in Example 6.9 and the chapter-opening example, we’ll draw a probability
tree (Figure 6.6). The notation is
C
= Has prostate cancer
CC
= Does not have prostate cancer
PT = Positive test result
NT = Negative test result
Starting with a man between 40 and 50 years old, we have the following probabilities
Prior
P(C) = .010
P(CC) = 1 - .010 = .990
Likelihood probabilities
False negative:
P(NT ƒ C) = .300
True positive:
P(PT ƒ C) = 1 - .300 = .700
False positive:
P(PT ƒ CC) = .135
True negative:
P(NT ƒ CC) = 1 - .135 = .865
FIGURE
6.6
Cancer
Probability Tree for Example 6.10
PSA test
P T |C .700
C .010
N T |C .300
P T | C c .135
C
c
.990
N T | C c .865
Joint probability
C and PT: (.010)(.700) = .0070
C and NT: (.010)(.300) = .0030
c
C and PT: (.990)(.135) = .1337
C c and NT: (.990)(.865) = .8564
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The tree allows you to determine the probability of obtaining a positive test result. It is
P(PT) = P(C and PT) + P(CC and PT) = .0070 + .1337 = .1407
We can now compute the probability that the man has prostate cancer given a positive
test result:
P(C ƒ PT) =
P(C and PT)
P(PT)
=
.0070
= .0498
.1407
The probability that he does not have prostate cancer is
P(CC ƒ PT) = 1 - P(C ƒ PT) = 1 - .0498 = .9502
We can repeat the process for the other age categories. Here are the results.
Probabilities Given a Positive PSA Test
Age
Has Prostate Cancer
Does Not Have Prostate Cancer
.0498
.1045
.2000
.3078
.9502
.8955
.8000
.6922
40–49
50–59
60–69
70 and older
The following table lists the proportion of each age category wherein the PSA test is
positive [P(PT)]
Age
40–49
50–59
60–79
70 and older
Proportion of
Tests That Are
Positive
Number of
Biopsies Performed
per Million
Number of
Cancers
Detected
Number of
Biopsies per
Cancer Detected
.1407
.1474
.1610
.1796
140,700
147,400
161,000
179,600
.0498(140,700) 7,007
.1045(147,400) 15,403
.2000(161,000) 32,200
.3078(179,600) 55,281
20.10
9.57
5.00
3.25
If we assume a cost of $1,000 per biopsy, the cost per cancer detected is $20,100 for
40 to 50, $9,570 for 50 to 60, $5,000 for 60 to 70, and $3,250 for over 70.
We have created an Excel spreadsheet to help you perform the calculations in
Example 6.10. Open the Excel Workbooks folder and select Medical screening.
There are three cells that you may alter. In cell B5, enter a new prior probability for
prostate cancer. Its complement will be calculated in cell B15. In cells D6 and D15, type
new values for the false-negative and false-positive rates, respectively. Excel will do the
rest. We will use this spreadsheet to demonstrate some terminology standard in medical
testing.
Terminology We will illustrate the terms using the probabilities calculated for the
40 to 50 age category.
The false-negative rate is .300. Its complement is the likelihood probability
P(PT ƒ C), called the sensitivity. It is equal to 1 .300 .700. Among men with prostate
cancer, this is the proportion of men who will get a positive test result.
The complement of the false-positive rate (.135) is P(NT ƒ CC), which is called the
specificity. This likelihood probability is 1 .135 .865
The posterior probability that someone has prostate cancer given a positive test
result [P(C ƒ PT) = .0498] is called the positive predictive value. Using Bayes’s Law, we can
compute the other three posterior probabilities.
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The probability that the patient does not have prostate cancer given a positive test
result is
P(CC ƒ PT) = .9502
The probability that the patient has prostate cancer given a negative test result is
P(C ƒ NT) = .0035
The probability that the patient does not have prostate cancer given a negative test
result:
P(CC ƒ NT) = .9965
This revised probability is called the negative predictive value.
Developing an Understanding of Probability Concepts
If you review the computations made previously, you’ll realize that the prior probabilities are as important as the probabilities associated with the test results (the likelihood
probabilities) in determining the posterior probabilities. The following table shows the
prior probabilities and the revised probabilities.
Age
Prior Probabilities
for Prostate Cancer
Posterior Probabilities
Given a Positive PSA Test
.010
.022
.046
.079
.0498
.1045
.2000
.3078
40–49
50–59
60–69
70 and older
As you can see, if the prior probability is low, then unless the screening test is quite
accurate, the revised probability will still be quite low.
To see the effects of different likelihood probabilities, suppose the PSA test is a
perfect predictor. In other words, the false-positive and false-negative rates are 0.
Figure 6.7 displays the probability tree.
FIGURE
6.7
Probability Tree for Example 6.10 with a Perfect Predictor Test
Cancer
PSA test
P T |C 1
C .010
N T |C 0
PT |C c 0
C
c
.990
NT |C c 1
Joint probability
C and PT: (.010)(1) = .010
C and NT: (.010)(0) = 0
c
C and PT: (.990)(0) = 0
C c and NT: (.990)(1) = .990
We find
P(PT ) = P(C and PT ) + P(CC and PT ) = .01 + 0 = .01
P(C ƒ PT ) =
P(C and PT )
P(PT )
=
.01
= 1.00
.01
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207
Now we calculate the probability of prostate cancer when the test is negative.
P(NT) = P(C and NT) + P(CC and NT) = 0 + .99 = .99
P(C ƒ NT) =
P(C and NT)
P(NT)
=
0
= 0
.99
Thus, if the test is a perfect predictor and a man has a positive test, then as expected the
probability that he has prostate cancer is 1.0. The probability that he does not have cancer when the test is negative is 0.
Now suppose that the test is always wrong; that is, the false-positive and falsenegative rates are 100%. The probability tree is shown in Figure 6.8.
FIGURE
6.8
Probability Tree for Example 6.10 with a Test That Is Always Wrong
Cancer
PSA test
P T |C 0
C .010
N T |C 1
PT |C
C
c
c
1
.990
NT |C c 0
Joint probability
C and PT: (.010)(0) = 0
C and NT: (.010)(1) = .010
c
C and PT: (.990)(1) = .990
C c and NT: (.990)(0) = 0
P(PT) = P(C and PT) + P(CC and PT) = 0 + .99 = .99
P(C ƒ PT) =
P(C and PT) 0
= 0
P(PT) .99
and
P(NT) = P(C and NT) + P(CC and NT) = .01 + 0 = .01
P(C ƒ NT) =
P(C and NT)
P(NT)
=
.01
= 1.00
.01
Notice we have another perfect predictor except that it is reversed. The probability of
prostate cancer given a positive test result is 0, but the probability becomes 1.00 when
the test is negative.
Finally we consider the situation when the set of likelihood probabilities are the
same. Figure 6.9 depicts the probability tree for a 40- to 50-year-old male and the probability of a positive test is (say) .3 and a the probability of a negative test is .7.
FIGURE
6.9
Cancer
Probability Tree for Example 6.10 with Identical
Likelihood Probabilities
PSA test
P T |C .3
C .010
N T |C .7
P T | C c .3
C
c
.990
N T | C c .7
Joint probability
C and PT: (.010)(.3) = .003
C and NT: (.010)(.7) = .007
c
C and PT: (.990)(.3) = .297
C c and NT: (.990)(.7) = .693
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P(PT) = P(C and PT) + P(CC and PT) = .003 + .297 = .300
P(C ƒ PT) =
P(C and PT)
P(PT)
=
.003
= .01
.300
and
P(NT) = P(C and NT) + P(CC and NT) = .007 + .693 = .700
P(C ƒ NT) =
P(C and NT)
P(NT)
= .007 + .700 = .01
As you can see, the posterior and prior probabilities are the same. That is, the PSA test
does not change the prior probabilities. Obviously, the test is useless.
We could have used any probability for the false-positive and false-negative rates,
including .5. If we had used .5, then one way of performing this PSA test is to flip a fair
coin. One side would be interpreted as positive and the other side as negative. It is clear
that such a test has no predictive power.
The exercises and Case 6.4 offer the probabilities for several other screening tests.
EXERCISES
6.71 Refer to Exercise 6.47. Determine P(A ƒ B).
6.72 Refer to Exercise 6.48. Find the following.
a.
b.
c.
d.
P(A ƒ B)
P(AC ƒ B)
P(A ƒ BC)
P(AC ƒ BC)
6.73 Refer to Example 6.9. An MBA applicant believes
Over a certain stretch of highway during a 1-year
period, suppose the probability of being involved in
a crash that results in at least one fatality is .01. It has
been estimated that 12% of the drivers on this highway drive while their BAC is greater than .09.
Determine the probability of a crash with at least
one fatality if a driver drives while legally intoxicated
(BAC greater than .09).
that the probability of scoring more than 650 on the
GMAT without the preparatory course is .95. What
is the probability of attaining that level after taking
the preparatory course?
6.76 Refer to Exercise 6.62. A randomly selected candi-
6.74 Refer to Exercise 6.58. The plant manager randomly
6.77 Bad gums may mean a bad heart. Researchers dis-
selects a molding from the early morning run and
discovers it is defective. What is the probability that
the foreman forgot to shut off the machine the previous night?
6.75 The U.S. National Highway Traffic Safety Admini-
stration gathers data concerning the causes of highway crashes where at least one fatality has occurred.
The following probabilities were determined from
the 1998 annual study (BAC is blood-alcohol content). (Source: Statistical Abstract of the United States,
2000, Table 1042.)
P(BAC 0 ƒ Crash with fatality) .616
P(BAC is between .01 and .09 ƒ Crash with fatality)
.300
P(BAC is greater than .09 ƒ Crash with fatality)
.084
date who took a CFA exam tells you that he has
passed the exam. What is the probability that he
took the CFA I exam?
covered that 85% of people who have suffered a
heart attack had periodontal disease, an inflammation of the gums. Only 29% of healthy people have
this disease. Suppose that in a certain community
heart attacks are quite rare, occurring with only 10%
probability. If someone has periodontal disease,
what is the probability that he or she will have a
heart attack?
6.78 Refer to Exercise 6.77. If 40% of the people in a
community will have a heart attack, what is the
probability that a person with periodontal disease
will have a heart attack?
6.79 Data from the Office on Smoking and Health,
Centers for Disease Control and Prevention, indicate that 40% of adults who did not finish high
school, 34% of high school graduates, 24% of adults
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PROBABILITY
who completed some college, and 14% of college
graduates smoke. Suppose that one individual is
selected at random, and it is discovered that the individual smokes. What is the probability that the individual is a college graduate? Use the probabilities in
Exercise 6.45 to calculate the probability that the
individual is a college graduate.
perfect. When the test is conducted on someone
whose kidney will be rejected, approximately one out
of five tests will be negative (i.e., the test is wrong).
When the test is conducted on a person whose kidney will not be rejected, 8% will show a positive test
result (i.e., another incorrect result). Physicians know
that in about 35% of kidney transplants the body
tries to reject the organ. Suppose that the test was
performed and the test is positive (indicating early
warning of rejection). What is the probability that
the body is attempting to reject the kidney?
6.80 Three airlines serve a small town in Ohio. Airline A
has 50% of all the scheduled flights, airline B has
30%, and airline C has the remaining 20%. Their
on-time rates are 80%, 65%, and 40%, respectively.
A plane has just left on time. What is the probability
that it was airline A?
209
6.83 The Rapid Test is used to determine whether some-
one has HIV (the virus that causes AIDS). The falsepositive and false-negative rates are .027 and .080,
respectively. A physician has just received the Rapid
Test report that his patient tested positive. Before
receiving the result, the physician assigned his
patient to the low-risk group (defined on the basis of
several variables) with only a 0.5% probability of
having HIV. What is the probability that the patient
actually has HIV?
6.81 Your favorite team is in the final playoffs. You have
assigned a probability of 60% that it will win the
championship. Past records indicate that when
teams win the championship, they win the first game
of the series 70% of the time. When they lose the
series, they win the first game 25% of the time. The
first game is over; your team has lost. What is the
probability that it will win the series?
6.84 What are the sensitivity, specificity, positive predic-
The following exercises are based on the Applications in Medical
Screening and Medical Insurance subsection.
tive value, and negative predictive value in the previous exercise?
6.82 Transplant operations have become routine. One
6.85 The Pap smear is the standard test for cervical
common transplant operation is for kidneys. The
most dangerous aspect of the procedure is the possibility that the body may reject the new organ. Several
new drugs are available for such circumstances, and
the earlier the drug is administered, the higher the
probability of averting rejection. The New England
Journal of Medicine recently reported the development of a new urine test to detect early warning signs
that the body is rejecting a transplanted kidney.
However, like most other tests, the new test is not
cancer. The false-positive rate is .636; the falsenegative rate is .180. Family history and age are
factors that must be considered when assigning a
probability of cervical cancer. Suppose that, after
obtaining a medical history, a physician determines that 2% of women of his patient’s age and
with similar family histories have cervical cancer.
Determine the effects a positive and a negative
Pap smear test have on the probability that the
patient has cervical cancer.
6 . 5 IDENTIFYING
THE
C OR R E C T M E T H O D
As we’ve previously pointed out, the emphasis in this book will be on identifying the
correct statistical technique to use. In Chapters 2 and 4, we showed how to summarize
data by first identifying the appropriate method to use. Although it is difficult to offer
strict rules on which probability method to use, we can still provide some general
guidelines.
In the examples and exercises in this text (and most other introductory statistics
books), the key issue is whether joint probabilities are provided or are required.
Joint Probabilities Are Given
In Section 6.2, we addressed problems where the joint probabilities were given. In these
problems, we can compute marginal probabilities by adding across rows and down
columns. We can use the joint and marginal probabilities to compute conditional
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probabilities, for which a formula is available. This allows us to determine whether the
events described by the table are independent or dependent.
We can also apply the addition rule to compute the probability that either of two
events occur.
Joint Probabilities Are Required
The previous section introduced three probability rules and probability trees. We need
to apply some or all of these rules in circumstances where one or more joint probabilities are required. We apply the multiplication rule (either by formula or through a
probability tree) to calculate the probability of intersections. In some problems, we’re
interested in adding these joint probabilities. We’re actually applying the addition rule
for mutually exclusive events here. We also frequently use the complement rule. In
addition, we can also calculate new conditional probabilities using Bayes’s Law.
CHAPTER SUM M ARY
The first step in assigning probability is to create an
exhaustive and mutually exclusive list of outcomes. The
second step is to use the classical, relative frequency, or
subjective approach and assign probability to the outcomes. A variety of methods are available to compute the
probability of other events. These methods include probability rules and trees.
An important application of these rules is Bayes’s
Law, which allows us to compute conditional probabilities
from other forms of probability.
IMPORTANT TERMS
Random experiment 176
Exhaustive 176
Mutually exclusive 176
Sample space 177
Classical approach 177
Relative frequency approach 178
Subjective approach 178
Event 178
Intersection 181
Joint probability 181
Marginal probability 183
Conditional probability 183
Independent events 185
Union 186
Complement 191
Complement rule 191
Multiplication rule 191
Addition rule 193
Bayes’s Law 199
Prior probability 200
Likelihood probability 200
Posterior probability 200
False-positive 203
False-negative 203
FORMULAS
Conditional probability
P(A ƒ B) =
P(A and B)
P(B)
Complement rule
Multiplication rule
P(A and B) = P(A ƒ B)P(B)
Addition rule
P(A or B) = P(A) + P(B) - P(A and B)
P(AC) = 1– P(A)
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211
CHAPTER EXERCISES
6.86 The following table lists the joint probabilities of
6.89 The effect of an antidepressant drug varies from
achieving grades of A and not achieving A’s in two
MBA courses.
person to person. Suppose that the drug is effective
on 80% of women and 65% of men. It is known that
66% of the people who take the drug are women.
What is the probability that the drug is effective?
Achieve a
Grade of A
in Marketing
Does Not
Achieve a
Grade of A
in Marketing
6.90 Refer to Exercise 6.89. Suppose that you are told
.053
.130
6.91 In a four-cylinder engine there are four spark plugs.
.237
.580
Achieve a grade of A
in statistics
Does not achieve a grade
of A in statistics
a. What is the probability that a student achieves a
grade of A in marketing?
b. What is the probability that a student achieves a
grade of A in marketing, given that he or she does
not achieve a grade of A in statistics?
c. Are achieving grades of A in marketing and statistics independent events? Explain.
6.87 A construction company has bid on two contracts.
The probability of winning contract A is .3. If the
company wins contract A, then the probability of
winning contract B is .4. If the company loses contract A, then the probability of winning contract B
decreases to .2. Find the probability of the following
events.
a. Winning both contracts
b. Winning exactly one contract
c. Winning at least one contract
that the drug is effective. What is the probability
that the drug taker is a man?
If any one of them malfunctions, the car will idle
roughly and power will be lost. Suppose that for a
certain brand of spark plugs the probability that a
spark plug will function properly after 5,000 miles is
.90. Assuming that the spark plugs operate independently, what is the probability that the car will idle
roughly after 5,000 miles?
6.92 A telemarketer sells magazine subscriptions over the
telephone. The probability of a busy signal or no
answer is 65%. If the telemarketer does make contact,
the probability of 0, 1, 2, or 3 magazine subscriptions
is .5, .25, .20, and .05, respectively. Find the probability that in one call she sells no magazines.
6.93 A statistics professor believes that there is a relation-
ship between the number of missed classes and the
grade on his midterm test. After examining his
records, he produced the following table of joint
probabilities.
6.88 Laser surgery to fix shortsightedness is becoming
more popular. However, for some people, a second
procedure is necessary. The following table lists the
joint probabilities of needing a second procedure
and whether the patient has a corrective lens with a
factor (diopter) of minus 8 or less.
First procedure
is successful
Second procedure
is required
Vision Corrective
Factor of
More Than
Minus 8
Vision Corrective
Factor of
Minus 8
or Less
.66
.15
.05
.14
a. Find the probability that a second procedure is
required.
b. Determine the probability that someone whose
corrective lens factor is minus 8 or less does not
require a second procedure.
c. Are the events independent? Explain your answer.
Student misses fewer
than 5 classes
Student misses 5 or
more classes
Student Fails
the Test
Student Passes
the Test
.02
.86
.09
.03
a. What is the pass rate on the midterm test?
b. What proportion of students who miss five or
more classes passes the midterm test?
c. What proportion of students who miss fewer
than five classes passes the midterm test?
d. Are the events independent?
6.94 In Canada, criminals are entitled to parole after
serving only one-third of their sentence. Virtually all
prisoners, with several exceptions including murderers, are released after serving two-thirds of their sentence. The government has proposed a new law that
would create a special category of inmates based on
whether they had committed crimes involving violence or drugs. Such criminals would be subject to
additional detention if the Correction Services
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judges them highly likely to reoffend. Currently,
27% of prisoners who are released commit another
crime within 2 years of release. Among those who
have reoffended, 41% would have been detained
under the new law, whereas 31% of those who have
not reoffended would have been detained.
a. What is the probability that a prisoner who
would have been detained under the new law
does commit another crime within 2 years?
b. What is the probability that a prisoner who
would not have been detained under the new law
does commit another crime within 2 years?
6.95 Casino Windsor conducts surveys to determine the
opinions of its customers. Among other questions,
respondents are asked to give their opinion about
“Your overall impression of Casino Windsor.” The
responses are
Excellent
Good
Average
Poor
In addition, the gender of the respondent is noted.
After analyzing the results, the following table of
joint probabilities was produced.
Rating
Excellent
Good
Average
Poor
Women
Men
.27
.14
.06
.03
.22
.10
.12
.06
a. What proportion of customers rate Casino
Windsor as excellent?
b. Determine the probability that a male customer
rates Casino Windsor as excellent.
c. Find the probability that a customer who rates
Casino Windsor as excellent is a man.
d. Are gender and rating independent? Explain
your answer.
6.96 A customer-service supervisor regularly conducts a
survey of customer satisfaction. The results of the
latest survey indicate that 8% of customers were not
satisfied with the service they received at their last
visit to the store. Of those who are not satisfied, only
22% return to the store within a year. Of those who
are satisfied, 64% return within a year. A customer
has just entered the store. In response to your question, he informs you that it is less than 1 year since
his last visit to the store. What is the probability that
he was satisfied with the service he received?
6.97 How does level of affluence affect health care? To
address one dimension of the problem, a group of
heart attack victims was selected. Each was categorized as a low-, medium-, or high-income earner.
Each was also categorized as having survived or
died. A demographer notes that in our society 21%
fall into the low-income group, 49% are in the
medium-income group, and 30% are in the highincome group. Furthermore, an analysis of heart
attack victims reveals that 12% of low-income
people, 9% of medium-income people, and 7% of
high-income people die of heart attacks. Find the
probability that a survivor of a heart attack is in
the low-income group.
6.98 A statistics professor and his wife are planning to
take a 2-week vacation in Hawaii, but they can’t
decide whether to spend 1 week on each of the
islands of Maui and Oahu, 2 weeks on Maui, or
2 weeks on Oahu. Placing their faith in random
chance, they insert two Maui brochures in one envelope, two Oahu brochures in a second envelope, and
one brochure from each island in a third envelope.
The professor’s wife will select one envelope at random, and their vacation schedule will be based on
the brochures of the islands so selected. After his
wife randomly selects an envelope, the professor
removes one brochure from the envelope (without
looking at the second brochure) and observes that it
is a Maui brochure. What is the probability that the
other brochure in the envelope is a Maui brochure?
(Proceed with caution: The problem is more difficult than it appears.)
6.99 The owner of an appliance store is interested in the
relationship between the price at which an item is
sold (regular or sale price) and the customer's decision on whether to purchase an extended warranty.
After analyzing her records, she produced the following joint probabilities.
Purchased
Did Not Purchase
Extended Warranty Extended Warranty
Regular price
Sale price
.21
.14
.57
.08
a. What is the probability that a customer who
bought an item at the regular price purchased the
extended warranty?
b. What proportion of customers buy an extended
warranty?
c. Are the events independent? Explain.
6.100 Researchers have developed statistical models
based on financial ratios that predict whether a
company will go bankrupt over the next 12 months.
In a test of one such model, the model correctly
predicted the bankruptcy of 85% of firms that did
in fact fail, and it correctly predicted nonbankruptcy for 74% of firms that did not fail. Suppose
that we expect 8% of the firms in a particular city
to fail over the next year. Suppose that the model
predicts bankruptcy for a firm that you own. What
is the probability that your firm will fail within the
next 12 months?
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PROBABILITY
b. What is the probability that the member felt that
at least one of these two issues was important?
6.101 A union’s executive conducted a survey of its mem-
CASE 6.1
6.102 In a class on probability, a statistics professor flips two
balanced coins. Both fall to the floor and roll under
his desk. A student in the first row informs the professor that he can see both coins. He reports that at least
one of them shows tails. What is the probability that
the other coin is also tails? (Beware the obvious.)
6.103 Refer to Exercise 6.102. Suppose the student
informs the professor that he can see only one coin
and it shows tails. What is the probability that the
other coin is also tails?
Let’s Make a Deal
© Michael Newman/PhotoEdit
bers to determine what the membership felt were
the important issues to be resolved during upcoming
negotiations with management. The results indicate
that 74% of members felt that job security was an
important issue, whereas 65% identified pension
benefits as an important issue. Of those who felt that
pension benefits were important, 60% also felt that
job security was an important issue. One member is
selected at random.
a. What is the probability that he or she felt that
both job security and pension benefits were
important?
213
number of years ago, there was
A
curtains, A, B, and C. Behind one of
a popular television game show
them is a brand new car worth
called Let’s Make a Deal. The
$20,000. Behind the other two cur-
host, Monty Hall, would randomly select
tains, the stage is empty. You decide to
contestants from the audience and, as
gamble and select curtain A. In an
the title suggests, he would make deals
attempt to make things more interest-
behind curtain C, what is the proba-
for prizes. Contestants would be given
ing, Monty then exposes an empty
bility that the car is behind curtain
relatively modest prizes and would then
stage by opening curtain C (he knows
A? What is the probability that the
be offered the opportunity to risk those
there is nothing behind curtain C). He
car is behind curtain B?
prizes to win better ones.
then offers you the free trip again if
Suppose that you are a contestant on
this show. Monty has just given you a
free trip touring toxic waste sites
around the country. He now offers you
2.
you quit now or, if you like, he will pro-
A? What is the probability that the
switch to curtain B). What do you do?
To help you answer that question, try
first answering these questions.
After Monty shows you what’s
bility that the car is behind curtain
your choice of curtain A or perhaps
for a gamble. On the stage are three
Before Monty shows you what’s
behind curtain C, what is the proba-
pose another deal (i.e., you can keep
a trade: Give up the trip in exchange
car is behind curtain B?
To Bunt or Not to Bunt, That
Is the Question
© AlBehrmanAP
CASE 6.2
1.
o sport generates as many sta-
N
Hal S. Stern, Vol. 1, Winter 1997) offers
tistics as baseball. Reporters,
baseball lovers another opportunity to
managers, and fans argue and
analyze numbers associated with the
discuss strategies on the basis of these
game. Table 1 lists the probabilities of
and the bases occupied. For example,
statistics. An article in Chance
scoring at least one run in situations
the probability of scoring at least one
(“A Statistician Reads the Sports Page,”
that are defined by the number of outs
run when there are no outs and a man
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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PROBABILITY
mother’s age. The probabilities are listed
Mother’s
Age
FalsePositive
Rate
FalseNegative
Rate
Under 30
30–34
35–37
Over 38
.04
.082
.178
.343
.376
.290
.269
.029
The probability that a baby has Down
syndrome is primarily a function of the
CASE 6.5
S
a.
45, and 49 determine the probabil-
here.
Age
25
30
35
40
45
49
For each of the ages 25, 30, 35, 40,
ity of Down syndrome if the mater-
Probability of
Down Syndrome
1/1300
1/900
1/350
1/100
1/25
1/12
Probability That at Least Two People in
the Same Room Have the Same Birthday
uppose that there are two people
are not July 1. If a third person now
in a room. The probability that they
enters the room, the probability that
share the same birthday (date, not
he or she has a different birthday from
nity serum screening produces a
positive result.
b.
Repeat for a negative result.
© Anna Jurkovska/Shutterstock
(Case 6.4 continued)
215
necessarily year) is 1/365, and the proba-
the first two people in the room is
that at least two have the same birth-
bility that they have different birthdays is
363/365. Thus, the probability that three
day.
364/365. To illustrate, suppose that you’re
people in a room having different birth-
Hint1: Calculate the probability that
in a room with one other person and that
days is (364/365)(363/365). You can con-
they don’t have the same birth-
your birthday is July 1. The probability
tinue this process for any number of
day.
that the other person does not have the
people.
same birthday is 364/365 because there
are 364 days in the year that
Hint 2: Excel users can employ the prod Find the number of people in a room so
uct function to calculate joint
that there is about a 50% probability
probabilities.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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7
© Comstock Images/Jupiterimages
RANDOM VARIABLES AND
DISCRETE PROBABILITY
DISTRIBUTIONS
7.1
Random Variables and Probability Distributions
7.2
Bivariate Distributions
7.3
(Optional) Applications in Finance:
Portfolio Diversification and Asset Allocation
7.4
Binomial Distribution
7.5
Poisson Distribution
DATA
Xm07-00
An investor has $100,000 to invest in the stock market. She is interested in developing a stock portfolio made up of stocks on the New York Stock Exchange (NYSE), the
Toronto Stock Exchange (TSX), and the NASDAQ. The stocks are Coca Cola and Disney
(NYSE), Barrick Gold (TSX), and Amazon (NASDAQ). However, she doesn’t know how much to invest
in each one. She wants to maximize her return, but she would also like to minimize the risk. She has
© Terry Vine/Blend Images/
Jupiterimages
Investing to Maximize Returns and Minimize Risk
computed the monthly returns for all four stocks during a 60-month period (January 2005 to
December 2009). After some consideration, she narrowed her choices down to the following three.
What should she do?
1. $25,000 in each stock
2. Coca Cola: $10,000, Disney: $20,000, Barrick Gold: $30,000, Amazon: $40,000
We will provide our
answer after we’ve
developed the necessary
tools in Section 7.3.
3. Coca Cola: $10,000, Disney: $50,000, Barrick Gold: $30,000, Amazon: $10,000
217
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CHAPTER 6
season. The results for the National
(Case 6.2 continued)
is on first base is .39. If the bases are
loaded with one out, then the probability of scoring any runs is .67.
1
TABLE
Probability of Scoring
Any Runs
Bases
Occupied
0 Outs 1 Out 2 Outs
Bases empty
First base
Second base
Third base
First base and
second base
First base and
third base
Second base and
third base
Bases loaded
.26
.39
.57
.72
.16
.26
.42
.55
.07
.13
.24
.28
4.
League are also shown in the article and
are similar.)
the runner advances.
Suppose that you are an American
Table 1 allows us to determine the best
League manager. The game is tied in the
strategy in a variety of circumstances.
middle innings of a game, and there is a
This case will concentrate on the strat-
runner on first base with no one out.
egy of the sacrifice bunt. The purpose of
Given the following probabilities of the
the sacrifice bunt is to sacrifice the bat-
four outcomes of a bunt for the batter
ter to move base runners to the next
at the plate, should you signal the bat-
base. It can be employed when there are
ter to sacrifice bunt?
fewer than two outs and men on base.
P(Outcome 1) .75
Ignoring the suicide squeeze, any of
P(Outcome 2) .10
four outcomes can occur:
.59
.76
.45
.61
.24
1.
(or runners) advances one base, and
.37
.74
.67
.37
.43
2.
P(Outcome 3) .10
P(Outcome 4) .05
The bunt is successful. The runner
the batter is out.
.83
.81
The batter is safe (hit or error), and
Assume for simplicity that after the hit
or error in outcome 4, there will be men
The batter is out but fails to
on first and second base and no one
advance the runner.
out.
(Probabilities are based on results from
the American League during the 1989
The batter bunts into a double play.
Should He Attempt to Steal a Base?
© AP Photo/AI Behrman
CASE 6.3
3.
R
ignore the relatively rare event wherein
strategy is to attempt to steal
the catcher throws the ball into center
second base. Historically the
field allowing the base runner to advance
probability of a successful steal of second
to third base.) Suppose there is a runner
base is approximately 68%. The probabil-
on first base. For each of the possible
whether it is advisable to have the runner
ity of being thrown out is 32%. (We’ll
number of outs (0, 1, or 2), determine
attempt to steal second base.
CASE 6.4
P
Maternal Serum Screening Test
for Down Syndrome
regnant women are screened
that their baby will be born with the
for a birth defect called Down
syndrome. The most common screen-
syndrome. Down syndrome
ing is maternal serum screening, a
© AP Photo/Javier Galeano
efer to Case 6.2. Another baseball
babies are mentally and physically
blood test that looks for markers in the
and false-negative rates vary accord-
challenged. Some mothers choose to
blood to indicate whether the birth
ing to the age of the mother.
abort the fetus when they are certain
defect may occur. The false-positive
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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CHAPTER 7
I N T RO D U C T I O N
7. 1
Page 218
I
n this chapter, we extend the concepts and techniques of probability introduced in
Chapter 6. We present random variables and probability distributions, which are
essential in the development of statistical inference.
Here is a brief glimpse into the wonderful world of statistical inference. Suppose
that you flip a coin 100 times and count the number of heads. The objective is to determine whether we can infer from the count that the coin is not balanced. It is reasonable
to believe that observing a large number of heads (say, 90) or a small number (say, 15)
would be a statistical indication of an unbalanced coin. However, where do we draw the
line? At 75 or 65 or 55? Without knowing the probability of the frequency of the number of heads from a balanced coin, we cannot draw any conclusions from the sample of
100 coin flips.
The concepts and techniques of probability introduced in this chapter will allow us
to calculate the probability we seek. As a first step, we introduce random variables and
probability distributions.
R A N D O M VA R I A B L E S
AND
P RO B A B I L I T Y D I S T R I B U T I O N S
Consider an experiment where we flip two balanced coins and observe the results. We
can represent the events as
Heads on the first coin and heads on the second coin
Heads on the first coin and tails on the second coin
Tails on the first coin and heads on the second coin
Tails on the first coin and tails on the second coin
However, we can list the events in a different way. Instead of defining the events by
describing the outcome of each coin, we can count the number of heads (or, if we wish,
the number of tails). Thus, the events are now
2 heads
1 heads
1 heads
0 heads
The number of heads is called the random variable. We often label the random variable X, and we’re interested in the probability of each value of X. Thus, in this illustration, the values of X are 0, 1, and 2.
Here is another example. In many parlor games as well as in the game of craps
played in casinos, the player tosses two dice. One way of listing the events is to describe
the number on the first die and the number on the second die as follows.
1, 1
2, 1
3, 1
4, 1
5, 1
6, 1
1, 2
2, 2
3, 2
4, 2
5, 2
6, 2
1, 3
2, 3
3, 3
4, 3
5, 3
6, 3
1, 4
2, 4
3, 4
4, 4
5, 4
6, 4
1, 5
2, 5
3, 5
4, 5
5, 5
6, 5
1, 6
2, 6
3, 6
4, 6
5, 6
6, 6
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
219
However, in almost all games, the player is primarily interested in the total.
Accordingly, we can list the totals of the two dice instead of the individual numbers.
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
If we define the random variable X as the total of the two dice, then X can equal 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, and 12.
Random Variable
A random variable is a function or rule that assigns a number to each outcome of an experiment.
In some experiments the outcomes are numbers. For example, when we observe
the return on an investment or measure the amount of time to assemble a computer, the
experiment produces events that are numbers. Simply stated, the value of a random
variable is a numerical event.
There are two types of random variables, discrete and continuous. A discrete random variable is one that can take on a countable number of values. For example, if we
define X as the number of heads observed in an experiment that flips a coin 10 times,
then the values of X are 0, 1, 2, . . . , 10. The variable X can assume a total of 11 values.
Obviously, we counted the number of values; hence, X is discrete.
A continuous random variable is one whose values are uncountable. An excellent
example of a continuous random variable is the amount of time to complete a task. For
example, let X ⫽ time to write a statistics exam in a university where the time limit is
3 hours and students cannot leave before 30 minutes. The smallest value of X is 30 minutes. If we attempt to count the number of values that X can take on, we need to identify the next value. Is it 30.1 minutes? 30.01 minutes? 30.001 minutes? None of these is
the second possible value of X because there exist numbers larger than 30 and smaller
than 30.001. It becomes clear that we cannot identify the second, or third, or any other
values of X (except for the largest value 180 minutes). Thus, we cannot count the number of values, and X is continuous.
A probability distribution is a table, formula, or graph that describes the values of a
random variable and the probability associated with these values. We will address discrete
probability distributions in the rest of this chapter and cover continuous distributions in
Chapter 8.
As we noted above, an uppercase letter will represent the name of the random variable, usually X. Its lowercase counterpart will represent the value of the random variable.
Thus, we represent the probability that the random variable X will equal x as
P(X = x)
or more simply
P(x)
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CHAPTER 7
Discrete Probability Distributions
The probabilities of the values of a discrete random variable may be derived by means
of probability tools such as tree diagrams or by applying one of the definitions of probability. However, two fundamental requirements apply as stated in the box.
Requirements for a Distribution of a Discrete Random Variable
1. 0 … P(x) … 1 for all x
2. a P(x) = 1
all x
where the random variable can assume values x and P(x) is the probability
that the random variable is equal to x.
These requirements are equivalent to the rules of probability provided in Chapter 6.
To illustrate, consider the following example.
EXAMPLE
7. 1
Probability Distribution of Persons per Household
The Statistical Abstract of the United States is published annually. It contains a wide
variety of information based on the census as well as other sources. The objective is
to provide information about a variety of different aspects of the lives of the country’s residents. One of the questions asks households to report the number of persons
living in the household. The following table summarizes the data. Develop the probability distribution of the random variable defined as the number of persons per
household.
Number of Persons
Number of Households (Millions)
1
2
3
4
5
6
7 or more
31.1
38.6
18.8
16.2
7.2
2.7
1.4
Total
116.0
Source: Statistical Abstract of the United States, 2009, Table 61.
SOLUTION
The probability of each value of X, the number of persons per household is computed
as the relative frequency. We divide the frequency for each value of X by the total number of households, producing the following probability distribution.
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
x
221
P (x)
31.1/116.0 ⫽ .268
38.6/116.0 ⫽ .333
18.8/116.0 ⫽ .162
16.2/116.0 ⫽ .140
7.2/116.0 ⫽ .062
2.7/116.0 ⫽ .023
1.4/116.0 ⫽ .012
1
2
3
4
5
6
7 or more
Total
1.000
As you can see, the requirements are satisfied. Each probability lies between 0 and 1,
and the total is 1.
We interpret the probabilities in the same way we did in Chapter 6. For example, if
we select one household at random, the probability that it has three persons is
P(3) = .162
We can also apply the addition rule for mutually exclusive events. (The values of X
are mutually exclusive; a household can have 1, 2, 3, 4, 5, 6, or 7 or more persons.) The
probability that a randomly selected household has four or more persons is
P(X Ú 4) = P(4) + P(5) + P(6) + P(7 or more)
= .140 + .062 + .023 + .012 = .237
In Example 7.1, we calculated the probabilities using census information about the
entire population. The next example illustrates the use of the techniques introduced in
Chapter 6 to develop a probability distribution.
EXAMPLE
7. 2
Probability Distribution of the Number of Sales
A mutual fund salesperson has arranged to call on three people tomorrow. Based on past
experience, the salesperson knows there is a 20% chance of closing a sale on each call.
Determine the probability distribution of the number of sales the salesperson will make.
SOLUTION
We can use the probability rules and trees introduced in Section 6.3. Figure 7.1 displays
the probability tree for this example. Let X ⫽ the number of sales.
FIGURE
7.1
Call 1
Call 2
S .2
S .2
SC .8
Call 3
Event
x
Probability
S .2
SSS
3
.008
SC .8
SSSC
2
.032
S .2
SSCS
2
.032
SC .8
SSCSC
1
.128
S .2
SCSS
2
.032
SC .8
SCSSC
1
.128
S S S
1
.128
SCSCSC
0
.512
SC .8
S .2
S .2
SC .8
SC .8
C C
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CHAPTER 7
The tree exhibits each of the eight possible outcomes and their probabilities. We
see that there is one outcome that represents no sales, and its probability is P(0) ⫽ .512.
There are three outcomes representing one sale, each with probability .128, so we add
these probabilities. Thus,
P(1) = .128 + .128 + .128 = 3(.128) = .384
The probability of two sales is computed similarly:
P(X) = 3(.032) = .096
There is one outcome where there are three sales:
P(3) = .008
The probability distribution of X is listed in Table 7.1.
TABLE
7.1
Probability Distribution of the Number of Sales in Example 7.2
x
P (x)
0
.512
1
.384
2
.096
3
.008
Probability Distributions and Populations
The importance of probability distributions derives from their use as representatives of
populations. In Example 7.1, the distribution provided us with information about the
population of numbers of persons per household. In Example 7.2, the population was
the number of sales made in three calls by the salesperson. And as we noted before, statistical inference deals with inference about populations.
Describing the Population/Probability Distribution
In Chapter 4, we showed how to calculate the mean, variance, and standard deviation of
a population. The formulas we provided were based on knowing the value of the random variable for each member of the population. For example, if we want to know the
mean and variance of annual income of all North American blue-collar workers, we
would record each of their incomes and use the formulas introduced in Chapter 4:
N
a Xi
m =
i=1
N
N
2
a (Xi - m)
s2 =
i=1
N
where X1 is the income of the first blue-collar worker, X2 is the second worker’s income,
and so on. It is likely that N equals several million. As you can appreciate, these formulas
are seldom used in practical applications because populations are so large. It is unlikely
that we would be able to record all the incomes in the population of North American
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223
blue-collar workers. However, probability distributions often represent populations.
Rather than record each of the many observations in a population, we list the values and
their associated probabilities as we did in deriving the probability distribution of the
number of persons per household in Example 7.1 and the number of successes in three
calls by the mutual fund salesperson. These can be used to compute the mean and variance of the population.
The population mean is the weighted average of all of its values. The weights are
the probabilities. This parameter is also called the expected value of X and is represented by E(X ).
Population Mean
E(X) = m = a x P(x)
all x
The population variance is calculated similarly. It is the weighted average of the
squared deviations from the mean.
Population Variance
V(X) = s2 = a (x - m)2P(x)
all x
There is a shortcut calculation that simplifies the calculations for the population
variance. This formula is not an approximation; it will yield the same value as the formula above.
Shortcut Calculation for Population Variance
V(X) = s2 = a x2P(x) - m2
all x
The standard deviation is defined as in Chapter 4.
Population Standard Deviation
s = 2s2
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CHAPTER 7
EXAMPLE
7. 3
Describing the Population of the Number of Persons
per Household
Find the mean, variance, and standard deviation for the population of the number of
persons per household Example 7.1.
SOLUTION
For this example, we will assume that the last category is exactly seven persons. The
mean of X is
E(X) = m = a xP(x) = 1P(1) + 2P(2) + 3P(3) + 4P(4) + 5P(5) + 6P(6) + 7P(7)
all x
= 1(.268) + 2(.333) + 3(.162) + 4(.140) + 5(.062) + 6(.023) + 7(.012)
= 2.512
Notice that the random variable can assume integer values only, yet the mean is 2.513.
The variance of X is
V(X) = s2 = a (x - m)2P(x)
all x
= (1 - 2.512)2(.268) + (2 - 2.512)2(.333) + (3 - 2.512)2(.162)
+ (4 - 2.512)2(.140) + (5 - 2.512)2(.062) + (6 - 2.512)2(.023)
+ (7 - 2.512)2(.012)
= 1.954
To demonstrate the shortcut method, we’ll use it to recompute the variance:
2
2
2
2
2
2
a x P(x) = 1 (.268) + 2 (.333) + 3 (.162) + 4 (.140) + 5 (.062)
all x
+ 62(.023) + 72(.012) = 8.264
and
m = 2.512
Thus,
s2 = a x2P(x) - m2 = 8.264 - (2.512)2 = 1.954
all x
The standard deviation is
s = 2s2 = 21.954 = 1.398
These parameters tell us that the mean and standard deviation of the number of
persons per household are 2.512 and 1.398, respectively.
Laws of Expected Value and Variance
As you will discover, we often create new variables that are functions of other random
variables. The formulas given in the next two boxes allow us to quickly determine the
expected value and variance of these new variables. In the notation used here, X is the
random variable and c is a constant.
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225
Laws of Expected Value
1. E(c) ⫽ c
2. E(X ⫹ c) ⫽ E(X) ⫹ c
3. E(cX) ⫽ cE(X)
Laws of Variance
1. V(c) ⫽ 0
2. V(X ⫹ c) ⫽ V(X)
3. V(c X) = c2V(X)
EXAMPLE
7. 4
Describing the Population of Monthly Profits
The monthly sales at a computer store have a mean of $25,000 and a standard deviation
of $4,000. Profits are calculated by multiplying sales by 30% and subtracting fixed costs
of $6,000. Find the mean and standard deviation of monthly profits.
SOLUTION
We can describe the relationship between profits and sales by the following equation:
Profit = .30(Sales) - 6,000
The expected or mean profit is
E(Profit) = E3.30(Sales) - 6,0004
Applying the second law of expected value, we produce
E(Profit) = E3.30(Sales)4 - 6,000
Applying law 3 yields
E(Profit) = .30E(Sales) - 6,000 = .30(25,000) - 6,000 = 1,500
Thus, the mean monthly profit is $1,500.
The variance is
V(Profit) = V 3.30(Sales) - 6,0004
The second law of variance states that
V(Profit) = V 3.30(Sales)4
and law 3 yields
V(Profit) = (.30)2V(Sales) = .09(4,000)2 = 1,440,000
Thus, the standard deviation of monthly profits is
sProfit = 21,440,000 = $1,200
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CHAPTER 7
EXERCISES
7.1 The number of accidents that occur on a busy
stretch of highway is a random variable.
a. What are the possible values of this random
variable?
b. Are the values countable? Explain.
c. Is there a finite number of values? Explain.
d. Is the random variable discrete or continuous?
Explain.
7.2 The distance a car travels on a tank of gasoline is a
random variable.
a. What are the possible values of this random
variable?
b. Are the values countable? Explain.
c. Is there a finite number of values? Explain.
d. Is the random variable discrete or continuous?
Explain.
a. Develop the probability distribution of X, the
number of color televisions per household.
b. Determine the following probabilities.
P(X … 2)
P(X ⬎ 2)
P(X Ú 4)
7.8 Using historical records, the personnel manager of a
plant has determined the probability distribution of
X, the number of employees absent per day. It is
x
0
P (x) .005
7.4 The mark on a statistics exam that consists of 100
multiple-choice questions is a random variable.
a. What are the possible values of this random variable?
b. Are the values countable? Explain.
c. Is there a finite number of values? Explain.
d. Is the random variable discrete or continuous?
Explain.
7.5 Determine whether each of the following is a valid
probability distribution.
a. x
P (x)
0
.1
1
.3
2
.4
3
.1
b. x
P (x)
5
.01
⫺6
.01
10
.01
0
.97
c. x
P (x)
14
.25
12
.46
⫺7
.04
13
.24
7.7 In a recent census the number of color televisions
per household was recorded
2
3
4
.310 .340 .220
5
.080
6
7
.019 .001
P(2 … X … 5)
P(X ⬎ 5)
P(X ⬍ 4)
b. Calculate the mean of the population.
c. Calculate the standard deviation of the population.
7.9 Second-year business students at many universities
are required to take 10 one-semester courses. The
number of courses that result in a grade of A is a discrete random variable. Suppose that each value of
this random variable has the same probability.
Determine the probability distribution.
7.10 The random variable X has the following probability
distribution.
x
P (x)
⫺3
2
6
8
.2
.3
.4
.1
7.11 An Internet pharmacy advertises that it will
ber of spots that turn up when a balanced die is
rolled. What is the probability distribution of X?
1
.025
3
Find the following probabilities.
a. P(X ⬎ 0)
b. P(X Ú 1)
c. P(X Ú 2)
d. P(2 … X … 5)
7.6 Let X be the random variable designating the num-
Number of color
televisions 0
2
a. Find the following probabilities.
7.3 The amount of money students earn on their sum-
mer jobs is a random variable.
a. What are the possible values of this random variable?
b. Are the values countable? Explain.
c. Is there a finite number of values? Explain.
d. Is the random variable discrete or continuous?
Explain.
1
4
5
Number of
households
(thousands)1,218 32,379 37,961 19,387 7,714 2,842
deliver the over-the-counter products that customers purchase in 3 to 6 days. The manager of
the company wanted to be more precise in its
advertising. Accordingly, she recorded the number of days it took to deliver to customers. From
the data, the following probability distribution
was developed.
Number of
days
0
1
2
3
4
5
6
7
8
Probability
0
0
.01
.04
.28
.42
.21
.02
.02
a. What is the probability that a delivery will be
made within the advertised 3- to 6-day period?
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
b. What is the probability that a delivery will be
late?
c. What is the probability that a delivery will be
early?
7.12 A gambler believes that a strategy called “doubling
up” is an effective way to gamble. The method
requires the gambler to double the stake after each
loss. Thus, if the initial bet is $1, after losing he
will double the bet until he wins. After a win, he
resorts back to a $1 bet. The result is that he will
net $1 for every win. The problem however, is that
he will eventually run out of money or bump up
against the table limit. Suppose that for a certain
game the probability of winning is .5 and that losing six in a row will result in bankrupting the gambler. Find the probability of losing six times in a
row.
7.13 The probability that a university graduate will be
offered no jobs within a month of graduation is estimated to be 5%. The probability of receiving one,
two, and three job offers has similarly been estimated to be 43%, 31%, and 21%, respectively.
Determine the following probabilities.
a. A graduate is offered fewer than two jobs.
b. A graduate is offered more than one job.
7.14 Use a probability tree to compute the probability
of the following events when flipping two fair
coins.
a. Heads on the first coin and heads on the second
coin
b. Heads on the first coin and tails on the second
coin
c. Tails on the first coin and heads on the second
coin
d. Tails on the first coin and tails on the second coin
7.15 Refer to Exercise 7.14. Find the following prob-
abilities.
a. No heads
b. One head
c. Two heads
d. At least one head
a. Find the mean and variance for the probability
distribution below.
b. Determine the probability distribution of Y
where Y ⫽ 5X.
c. Use the probability distribution in part (b) to
compute the mean and variance of Y.
d. Use the laws of expected value and variance to
find the expected value and variance of Y from
the parameters of X.
7.19 We are given the following probability distribution.
x
0
1
2
3
P (x)
.4
.3
.2
.1
a. Calculate the mean, variance, and standard
deviation.
b. Suppose that Y ⫽ 3X ⫹ 2. For each value of X,
determine the value of Y. What is the probability
distribution of Y?
c. Calculate the mean, variance, and standard deviation from the probability distribution of Y.
d. Use the laws of expected value and variance to
calculate the mean, variance, and standard deviation of Y from the mean, variance, and standard deviation of X. Compare your answers in
parts (c) and (d). Are they the same (except for
rounding)?
7.20 The number of pizzas delivered to university stu-
dents each month is a random variable with the
following probability distribution.
x
0
1
2
3
P (X)
.1
.3
.4
.2
a. Find the probability that a student has received
delivery of two or more pizzas this month.
b. Determine the mean and variance of the number
of pizzas delivered to students each month.
7.21 Refer to Exercise 7.20. If the pizzeria makes a profit
of $3 per pizza, determine the mean and variance of
the profits per student.
7.22 After watching a number of children playing games
7.16 Draw a probability tree to describe the flipping of
three fair coins.
7.17 Refer to Exercise 7.16. Find the following
probabilities.
a. Two heads
b. One head
c. At least one head
d. At least two heads
at a video arcade, a statistics practitioner estimated
the following probability distribution of X, the number of games per visit.
x
7.18 The random variable X has the following distribution.
x
⫺2
5
7
8
P (x)
.59
.15
.25
.01
1
2
3
4
5
6
7
P (x) .05
.15
.15
.25
.20
.10
.10
a. What is the probability that a child will play
more than four games?
b. What is the probability that a child will play at
least two games?
7.23 Refer to Exercise 7.22. Determine the mean and
variance of the number of games played.
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7.24 Refer to Exercise 7.23. Suppose that each game costs
the player 25 cents. Use the laws of expected value
and variance to determine the expected value and
variance of the amount of money the arcade takes in.
7.25 Refer to Exercise 7.22.
a. Determine the probability distribution of the
amount of money the arcade takes in per child.
b. Use the probability distribution to calculate the
mean and variance of the amount of money the
arcade takes in.
c. Compare the answers in part (b) with those of
Exercise 7.24. Are they identical (except for
rounding errors)?
7.26 A survey of Amazon.com shoppers reveals the fol-
lowing probability distribution of the number of
books purchased per hit.
x
0
P (x) .35
1
2
3
4
5
6
7
.25
.20
.08
.06
.03
.02
.01
a. What is the probability that an
visitor will buy four books?
b. What is the probability that an
visitor will buy eight books?
c. What is the probability that an
visitor will not buy any books?
d. What is the probability that an
visitor will buy at least one book?
of flu and colds. A 6-month study was undertaken to
determine whether the remedy works. From this
study, the following probability distribution of the
number of respiratory infections per year (X) for
echinacea users was produced.
x
1
2
3
4
.31
.17
.06
.01
Find the following probabilities.
a. An echinacea user has more than one infection
per year.
b. An echinacea user has no infections per year.
c. An echinacea user has between one and three
(inclusive) infections per year.
7.30 A shopping mall estimates the probability distribu-
tion of the number of stores mall customers actually
enter, as shown in the table.
x
Amazon.com
0
P (x) .45
0
P (x) .04
1
2
3
4
5
6
.19
.22
.28
.12
.09
.06
Amazon.com
Find the mean and standard deviation of the number
of stores entered.
Amazon.com
7.31 Refer to Exercise 7.30. Suppose that, on average,
Amazon.com
customers spend 10 minutes in each store they enter.
Find the mean and standard deviation of the total
amount of time customers spend in stores.
7.27 A university librarian produced the following proba-
7.32 When parking a car in a downtown parking lot, dri-
bility distribution of the number of times a student
walks into the library over the period of a semester.
x
0
P (x)
5
10
15
.22 .29 .12
20
25 30
40
50
75 100
.09 .08 .05 .04 .04 .03 .03 .01
Find the following probabilities.
a. P(X Ú 20)
b. P(X ⫽ 60)
c. P(X ⬎ 50)
d. P(X ⬎ 100)
0
.04
1
2
3
.09 .19 .21
1
P (x) .24
2
3
4
5
6
7
8
.18
.13
.10
.07
.04
.04
.20
7.33 Refer to Exercise 7.32. The cost of parking is $2.50
country skiers participate in their sport, a sportswriter created the following probability distribution
for X ⫽ number of times per year cross-country
skiers ski.
P (x)
x
Find the mean and standard deviation of the number
of hours cars are parked in the lot.
7.28 After analyzing the frequency with which cross-
x
vers pay according to the number of hours or parts
thereof. The probability distribution of the number of
hours cars are parked has been estimated as follows.
4
5
.16
.12
6
7
8
.08 .06 .05
Find the following.
a. P(3)
b. P(X Ú 5)
c. P(5 … X … 7)
7.29 The natural remedy echinacea is reputed to boost
the immune system, which will reduce the number
per hour. Calculate the mean and standard deviation
of the amount of revenue each car generates.
7.34 You have been given the choice of receiving $500 in
cash or receiving a gold coin that has a face value of
$100. However, the actual value of the gold coin
depends on its gold content. You are told that the
coin has a 40% probability of being worth $400, a
30% probability of being worth $900, and a 30%
probability of being worth its face value. Basing your
decision on expected value, should you choose the
coin?
7.35 The manager of a bookstore recorded the number of
customers who arrive at a checkout counter every
5 minutes from which the following distribution was
calculated. Calculate the mean and standard deviation of the random variable.
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
x
0
P (x) .10
1
2
3
4
.20
.25
.25
.20
Compute the mean and variance of the number of
pages per fax.
7.36 The owner of a small firm has just purchased a per-
sonal computer, which she expects will serve her for
the next 2 years. The owner has been told that she
“must” buy a surge suppressor to provide protection
for her new hardware against possible surges or variations in the electrical current, which have the
capacity to damage the computer. The amount of
damage to the computer depends on the strength of
the surge. It has been estimated that there is a 1%
chance of incurring $400 damage, a 2% chance of
incurring $200 damage, and 10% chance of $100
damage. An inexpensive suppressor, which would
provide protection for only one surge can be purchased. How much should the owner be willing to
pay if she makes decisions on the basis of expected
value?
7.37 It cost one dollar to buy a lottery ticket, which has
five prizes. The prizes and the probability that a
player wins the prize are listed here. Calculate the
expected value of the payoff.
Prize ($)
1 million
200,000
50,000
Probability
1/10 million
1/1 million
1/500,000
Prize ($)
10,000
1,000
Probability
1/50,000
1/10,000
7.38 After an analysis of incoming faxes the manager of
an accounting firm determined the probability distribution of the number of pages per facsimile as
follows:
x
P (x)
229
1
2
3
4
5
6
7
.05
.12
.20
.30
.15
.10
.08
7.39 Refer to Exercise 7.38. Further analysis by the man-
ager revealed that the cost of processing each page
of a fax is $.25. Determine the mean and variance of
the cost per fax.
7.40 To examine the effectiveness of its four annual adver-
tising promotions, a mail-order company has sent a
questionnaire to each of its customers, asking how
many of the previous year’s promotions prompted
orders that would not otherwise have been made.
The table lists the probabilities that were derived
from the questionnaire, where X is the random variable representing the number of promotions that
prompted orders. If we assume that overall customer
behavior next year will be the same as last year, what
is the expected number of promotions that each customer will take advantage of next year by ordering
goods that otherwise would not be purchased?
x
P (x)
0
1
2
3
4
.10
.25
.40
.20
.05
7.41 Refer to Exercise 7.40. A previous analysis of histor-
ical records found that the mean value of orders for
promotional goods is $20, with the company earning a gross profit of 20% on each order. Calculate
the expected value of the profit contribution next
year.
7.42 Refer to Exercises 7.40 and 7.41. The fixed cost of
conducting the four promotions is estimated to be
$15,000, with a variable cost of $3.00 per customer
for mailing and handling costs. How large a customer base does the company need to cover the cost
of promotions?
7. 2 B I VA R I AT E D I S T R I B U T I O N S
Thus far, we have dealt with the distribution of a single variable. However, there are circumstances where we need to know about the relationship between two variables.
Recall that we have addressed this problem statistically in Chapter 3 by drawing the
scatter diagram and in Chapter 4 by calculating the covariance and the coefficient of
correlation. In this section, we present the bivariate distribution, which provides
probabilities of combinations of two variables. Incidentally, when we need to distinguish between the bivariate distributions and the distributions of one variable, we’ll
refer to the latter as univariate distributions.
The joint probability that two variables will assume the values x and y is denoted
P(x, y). A bivariate (or joint) probability distribution of X and Y is a table or formula that
lists the joint probabilities for all pairs of values of x and y. As was the case with univariate distributions, the joint probability must satisfy two requirements.
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Requirements for a Discrete Bivariate Distribution
1. 0 … P(x, y) … 1 for all pairs of values (x, y)
2. a a P(x, y) = 1
all x all y
EXAMPLE
7. 5
Bivariate Distribution of the Number of House Sales
Xavier and Yvette are real estate agents. Let X denote the number of houses that Xavier
will sell in a month and let Y denote the number of houses Yvette will sell in a month.
An analysis of their past monthly performances has the following joint probabilities.
Bivariate Probability Distribution
X
0
1
2
Y
0
1
2
.12
.21
.07
.42
.06
.02
.06
.03
.01
We interpret these joint probabilities in the same way we did in Chapter 6. For
example, the probability that Xavier sells 0 houses and Yvette sells 1 house in the month
is P(0, 1) ⫽ .21.
Marginal Probabilities
As we did in Chapter 6, we can calculate the marginal probabilities by summing across
rows or down columns.
Marginal Probability Distribution of X in Example 7.5
P(X = 0) = P(0, 0) + P(0, 1) + P(0, 2) = .12 + .21 + .07 = .4
P(X = 1) = P(1, 0) + P(1, 1) + P(1, 2) = .42 + .06 + .02 = .5
P(X = 2) = P(2, 0) + P(2, 1) + P(2, 2) = .06 + .03 + .01 = .1
The marginal probability distribution of X is
x
P (x)
0
1
2
.4
.5
.1
Marginal Probability Distribution of Y in Example 7.5
P(Y = 0) = P(0, 0) + P(1, 0) + P(2, 0) = .12 + .42 + .06 = .6
P(Y = 1) = P(0, 1) + P(1, 1) + P(2, 1) = .21 + .06 + .03 = .3
P(Y = 2) = P(0, 2) + P(1, 2) + P(2, 2) = .07 + .02 + .01 = .1
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231
The marginal probability distribution of Y is
y
P (y)
0
1
2
.6
.3
.1
Notice that both marginal probability distributions meet the requirements; the probabilities are between 0 and 1, and they add to 1.
Describing the Bivariate Distribution
As we did with the univariate distribution, we often describe the bivariate distribution
by computing the mean, variance, and standard deviation of each variable. We do so by
utilizing the marginal probabilities.
Expected Value, Variance, and Standard Deviation of X in Example 7.5
E(X) = mX = a x P(x) = 0(.4) + 1(.5) + 2(.1) = .7
V(X) = s2X = a (x - mX)2P(x) = (0 - .7)2(.4) + (1 - .7)2(.5) + (2 - .7)2(.1) = .41
sX = 2s2X = 2.41 = .64
Expected Value, Variance, and Standard Deviation of Y in Example 7.5
E(Y ) = mY = a y P(y) = 0(.6) + 1(.3) + 2(.1) = .5
V(Y) = s2Y = a (y - mY)2P(y) = (0 - .5)2(.6) + (1 - .5)2(.3) + (2 - .5)2(.1) = .45
sY = 2s2Y = 2.45 = .67
There are two more parameters we can and need to compute. Both deal with the relationship between the two variables. They are the covariance and the coefficient of correlation. Recall that both were introduced in Chapter 4, where the formulas were based
on the assumption that we knew each of the N observations of the population. In this
chapter, we compute parameters like the covariance and the coefficient of correlation
from the bivariate distribution.
Covariance
The covariance of two discrete variables is defined as
COV(X,Y) = sxy = a a (x - mX)(y - mY)P(x, y)
all x all y
Notice that we multiply the deviations from the mean for both X and Y and then multiply by the joint probability.
The calculations are simplified by the following shortcut method.
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CHAPTER 7
Shortcut Calculation for Covariance
COV(X,Y) = sxy = a a xyP(x, y) - mX mY
all x all y
The coefficient of correlation is calculated in the same way as in Chapter 4.
Coefficient of Correlation
r =
EXAMPLE
7. 6
sxy
sxsy
Describing the Bivariate Distribution
Compute the covariance and the coefficient of correlation between the numbers of
houses sold by the two agents in Example 7.5.
SOLUTION
We start by computing the covariance.
sxy = a a (x - mX)(y - mY)P(x, y)
all x all y
= (0 - .7)(0 - .5)(.12) + (1 - .7)(0 - .5)(.42) + (2 - .7)(0 - .5)(.06)
+ (0 - .7)(1 - .5)(.21) + (1 - .7)(1 - .5)(.06) + (2 - .7)(1 - .5)(03)
+ (0 - .7)(2 - .5)(.07) + (1 - .7)(2 - .5)(.02) + (2 - .7)(2 - .5)(.01)
= - .15
As we did with the shortcut method for the variance, we’ll recalculate the covariance using its shortcut method.
a a xyP(x, y) = (0)(0)(.12) + (1)(0)(.42) + (2)(0)(.06)
+ (0)(1)(.21) + (1)(1)(.06) + (2)(1)(.03)
+ (0)(2)(.07) + (1)(2)(.02) + (2)(2)(.01)
= .2
all x all y
Using the expected values computed above we find
sxy = a a xyP(x, y) - mX mY = .2 - (.7)(.5) = - .15
all x all y
We also computed the standard deviations above. Thus, the coefficient of
correlation is
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r =
sxy
sX sY
=
233
-.15
= - .35
(.64)(.67)
There is a weak negative relationship between the two variables: the number of
houses Xavier will sell in a month (X ) and the number of houses Yvette will sell in a
month (Y ).
Sum of Two Variables
The bivariate distribution allows us to develop the probability distribution of any combination of the two variables. Of particular interest to us is the sum of two variables.
The analysis of this type of distribution leads to an important statistical application in
finance, which we present in the next section.
To see how to develop the probability distribution of the sum of two variables from
their bivariate distribution, return to Example 7.5. The sum of the two variables X and Y
is the total number of houses sold per month. The possible values of X ⫹ Y are 0, 1, 2, 3,
and 4. The probability that X ⫹ Y ⫽ 2, for example, is obtained by summing the joint
probabilities of all pairs of values of X and Y that sum to 2:
P(X + Y = 2) = P(0,2) + P(1,1) + P(2,0) = .07 + .06 + .06 = .19
We calculate the probabilities of the other values of X ⫹ Y similarly, producing the
following table.
Probability Distribution of X ⫹ Y in Example 7.5
x⫹y
P (x ⫹ y)
0
1
2
3
4
.12
.63
.19
.05
.01
We can compute the expected value, variance, and standard deviation of X ⫹ Y in
the usual way.
E(X + Y) = 0(.12) + 1(.63) + 2(.19) + 3(.05) + 4(.01) = 1.2
V(X + Y) = s2X + Y = (0 - 1.2)2(.12) + (1 - 1.2)2(.63) + (2 - 1.2)2(.19)
+ (3 - 1.2)2(.05) + (4 - 1.2)2(.01)
= .56
sX + Y = 2.56 = .75
We can derive a number of laws that enable us to compute the expected value and
variance of the sum of two variables.
Laws of Expected Value and Variance of the Sum of Two Variables
1. E(X + Y ) = E(X ) + E(Y )
2. V(X + Y ) = V(X ) + V(Y ) + 2COV(X, Y )
If X and Y are independent, COV(X, Y ) = 0 and thus V(X + Y )
= V(X ) + V(Y )
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EXAMPLE
Describing the Population of the Total Number
of House Sales
7. 7
Use the rules of expected value and variance of the sum of two variables to calculate the
mean and variance of the total number of houses sold per month in Example 7.5.
SOLUTION
Using law 1 we compute the expected value of X ⫹ Y:
E(X + Y ) = E(X ) + E(Y ) = .7 + .5 = 1.2
which is the same value we produced directly from the probability distribution of X ⫹ Y.
We apply law 3 to determine the variance:
V1X + Y2 = V1X2 + V1Y2 + 2COV1X,Y2 = .41 + .45 + 21- .152 = .56
This is the same value we obtained from the probability distribution of X ⫹ Y.
We will encounter several applications where we need the laws of expected
value and variance for the sum of two variables. Additionally, we will demonstrate
an important application in operations management where we need the formulas
for the expected value and variance of the sum of more than two variables. See
Exercises 7.57–7.60.
EXERCISES
7.43 The following table lists the bivariate distribution of
7.47 The bivariate distribution of X and Y is described here.
X and Y.
x
x
a.
b.
c.
d.
y
1
2
1
2
.5
.1
.1
.3
Find the marginal probability distribution of X.
Find the marginal probability distribution of Y.
Compute the mean and variance of X.
Compute the mean and variance of Y.
7.44 Refer to Exercise 7.43. Compute the covariance and
the coefficient of correlation.
7.45 Refer to Exercise 7.43. Use the laws of expected
value and variance of the sum of two variables to
compute the mean and variance of X ⫹ Y.
7.46 Refer to Exercise 7.43.
a. Determine the distribution of X ⫹ Y.
b. Determine the mean and variance of X ⫹ Y.
c. Does your answer to part (b) equal the answer to
Exercise 7.45?
a.
b.
c.
d.
y
1
2
1
2
.28
.12
.42
.18
Find the marginal probability distribution of X.
Find the marginal probability distribution of Y.
Compute the mean and variance of X.
Compute the mean and variance of Y.
7.48 Refer to Exercise 7.47. Compute the covariance and
the coefficient of correlation.
7.49 Refer to Exercise 7.47. Use the laws of expected
value and variance of the sum of two variables to
compute the mean and variance of X ⫹ Y.
7.50 Refer to Exercise 7.47.
a. Determine the distribution of X ⫹ Y.
b. Determine the mean and variance of X ⫹ Y.
c. Does your answer to part (b) equal the answer to
Exercise 7.49?
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
7.51 The joint probability distribution of X and Y is
shown in the following table.
x
y
1
2
3
1
2
.42
.28
.12
.08
.06
.04
c. Compute the mean and variance of the number
of refrigerators sold daily.
d. Compute the mean and variance of the number
of stoves sold daily.
e. Compute the covariance and the coefficient of
correlation.
7.55 Canadians who visit the United States often buy
a. Determine the marginal distributions of X and Y.
b. Compute the covariance and coefficient of correlation between X and Y.
c. Develop the probability distribution of X ⫹ Y.
7.52 The following distributions of X and of Y have been
developed. If X and Y are independent, determine
the joint probability distribution of X and Y.
x
0
1
2
y
1
2
p(x)
.6
.3
.1
p(y)
.7
.3
liquor and cigarettes, which are much cheaper in the
United States. However, there are limitations.
Canadians visiting in the United States for more
than 2 days are allowed to bring into Canada one
bottle of liquor and one carton of cigarettes. A
Canada customs agent has produced the following
joint probability distribution of the number of bottles of liquor and the number of cartons of cigarettes
imported by Canadians who have visited the United
States for 2 or more days.
Bottles of Liquor
7.53 The distributions of X and of Y are described here.
If X and Y are independent, determine the joint
probability distribution of X and Y.
x
0
1
y
1
2
3
P (x)
.2
.8
P (y)
.2
.4
.4
7.54 After analyzing several months of sales data, the
owner of an appliance store produced the following
joint probability distribution of the number of
refrigerators and stoves sold daily.
Refrigerators
Stoves
0
1
2
235
0
1
2
.08
.09
.05
.14
.17
.18
.12
.13
.04
Cartons of Cigarettes
0
1
0
1
.63
.09
.18
.10
a. Find the marginal probability distribution of the
number of bottles imported.
b. Find the marginal probability distribution of the
number of cigarette cartons imported.
c. Compute the mean and variance of the number
of bottles imported.
d. Compute the mean and variance of the number
of cigarette cartons imported.
e. Compute the covariance and the coefficient of
correlation.
7.56 Refer to Exercise 7.54. Find the following condi-
a. Find the marginal probability distribution of the
number of refrigerators sold daily.
b. Find the marginal probability distribution of the
number of stoves sold daily.
tional probabilities.
a. P(1 refrigerator ƒ 0 stoves)
b. P(0 stoves ƒ 1 refrigerator)
c. P(2 refrigerators ƒ 2 stoves)
APPLICATIONS in OPERATIONS MANAGEMENT
PERT/CPM
The Project Evaluation and Review Technique (PPERT) and the Critical Path Method
© Susan Van Etten
CPM) are related management-science techniques that help operations man(C
agers control the activities and the amount of time it takes to complete a project.
Both techniques are based on the order in which the activities must be performed. For example, in building a house the excavation of the foundation must
precede the pouring of the foundation, which in turn precedes the framing. A path
(Continued)
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CHAPTER 7
is defined as a sequence of related activities that leads from the starting point to the completion of
a project. In most projects, there are several paths with differing amounts of time needed for their
completion. The longest path is called the critical path because any delay in the activities along
this path will result in a delay in the completion of the project. In some versions of PERT/CPM, the
activity completion times are fixed and the chief task of the operations manager is to determine
the critical path. In other versions, each activity’s completion time is considered to be a random
variable, where the mean and variance can be estimated. By extending the laws of expected value
and variance for the sum of two variables to more than two variables, we produce the following,
where X1, X2, . . . , Xk are the times for the completion of activities 1, 2, . . . , k, respectively. These
times are independent random variables.
Laws of Expected Value and Variance for the Sum of More than Two Independent Variables
1. E(X1 + X2 + Á + Xk) = E(X1) + E(X2) + Á + E(XK)
2. V (X1 + X2 + Á + Xk) = V (X1) + V (X2) + Á + V (XK)
Using these laws, we can then produce the expected value and variance for the complete project.
Exercises 7.57–7.60 address this problem.
7.57 There are four activities along the critical path for a project. The expected values
and variances of the completion times of the activities are listed here. Determine
the expected value and variance of the completion time of the project.
Activity
Expected Completion
Time (Days)
Variance
1
2
3
4
18
12
27
8
8
5
6
2
7.58 The operations manager of a large plant wishes to overhaul a machine. After
conducting a PERT/CPM analysis he has developed the following critical path.
1.
2.
3.
4.
5.
Disassemble machine
Determine parts that need replacing
Find needed parts in inventory
Reassemble machine
Test machine
He has estimated the mean (in minutes) and variances of the completion times as
follows.
Activity
Mean
Variance
1
2
3
4
5
35
20
20
50
20
8
5
4
12
2
Determine the mean and variance of the completion time of the project.
7.59 In preparing to launch a new product, a marketing manager has determined the
critical path for her department. The activities and the mean and variance of
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
237
the completion time for each activity along the critical path are shown in the
accompanying table. Determine the mean and variance of the completion time
of the project.
Activity
Expected Completion
Time (Days)
Variance
8
14
5
3
30
30
10
2
5
1
1
8
10
3
Develop survey questionnaire
Pretest the questionnaire
Revise the questionnaire
Hire survey company
Conduct survey
Analyze data
Prepare report
7.60 A professor of business statistics is about to begin work on a new research project.
Because his time is quite limited, he has developed a PERT/CPM critical path,
which consists of the following activities:
1.
2.
3.
4.
5.
6.
Conduct a search for relevant research articles.
Write a proposal for a research grant.
Perform the analysis.
Write the article and send to journal.
Wait for reviews.
Revise on the basis of the reviews and resubmit.
The mean (in days) and variance of the completion times are as follows
Activity
Mean
Variance
1
2
3
4
5
6
10
3
30
5
100
20
9
0
100
1
400
64
Compute the mean and variance of the completion time of the entire project.
7. 3
(O P T I O N A L ) A PPL I C AT I O N S I N F I N A N C E : P ORT F O L I O
D I V E R S I F I C AT I O N A N D A S S E T A L LO C AT I O N
In this section we introduce an important application in finance that is based on the
previous section.
In Chapter 3 (page 51), we described how the variance or standard deviation can
be used to measure the risk associated with an investment. Most investors tend to be
risk averse, which means that they prefer to have lower risk associated with their
investments. One of the ways in which financial analysts lower the risk that is associated with the stock market is through diversification. This strategy was first mathematically developed by Harry Markowitz in 1952. His model paved the way for the
development of modern portfolio theory (MPT), which is the concept underlying
mutual funds (see page 181).
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CHAPTER 7
To illustrate the basics of portfolio diversification, consider an investor who forms
a portfolio, consisting of only two stocks, by investing $4,000 in one stock and $6,000 in
a second stock. Suppose that the results after 1 year are as listed here. (We’ve previously
defined return on investment. See Applications in Finance: Return on Investment on
page 52.)
One-Year Results
Stock
Initial
Investment ($)
Value of Investment
After One Year ($)
1
2
4,000
6,000
5,000
5,400
Total
10,000
10,400
Rate of Return
on Investment
R1 ⫽ .25 (25%)
R2 ⫽ ⫺.10 (⫺10%)
Rp ⫽ .04 (4%)
Another way of calculating the portfolio return Rp is to compute the weighted average of the individual stock returns R1 and R2, where the weights w1 and w2 are the proportions of the initial $10,000 invested in stocks 1 and 2, respectively. In this illustration,
w1 ⫽ .4 and w2 ⫽ .6. (Note that w1 and w2 must always sum to 1 because the two stocks
constitute the entire portfolio.) The weighted average of the two returns is
Rp = w1R1 + w2R2
= (.4)(.25) + (.6)(-.10) = .04
This is how portfolio returns are calculated. However, when the initial investments are
made, the investor does not know what the returns will be. In fact, the returns are random variables. We are interested in determining the expected value and variance of the
portfolio. The formulas in the box were derived from the laws of expected value and
variance introduced in the two previous sections.
Mean and Variance of a Portfolio of Two Stocks
E(Rp) = w1E(R1) + w2E(R2)
V(Rp) = w21V(R1) + w22V(R2) + 2w1w2COV(R1,R2)
= w21s21 + w22s22 + 2w1w2rs1s2
where w1 and w2 are the proportions or weights of investments 1 and 2,
E(R1) and E(R2) are their expected values, ␴1 and ␴2 are their standard deviations, COV(R1,R2) is the covariance, and ␳ is the coefficient of correlation.
(Recall that r =
EXAMPLE
7. 8
COV(R1,R2)
s1s2
, which means that COV(R1,R2) ⫽ ␴1 ␴2.)
Describing the Population of the Returns on a Portfolio
An investor has decided to form a portfolio by putting 25% of his money into
McDonald’s stock and 75% into Cisco Systems stock. The investor assumes that the
expected returns will be 8% and 15%, respectively, and that the standard deviations will
be 12% and 22%, respectively.
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
239
a. Find the expected return on the portfolio.
b. Compute the standard deviation of the returns on the portfolio assuming that
i. the two stocks’ returns are perfectly positively correlated.
ii. the coefficient of correlation is .5.
iii. the two stocks’ returns are uncorrelated.
SOLUTION
a. The expected values of the two stocks are
E(R1) = .08
and
E(R2) = .15
The weights are w1 ⫽ .25 and w2 ⫽ .75.
Thus,
E(Rp) = w1E(R1) + w2E(R2) = .25(.08) + .75(.15) = .1325
b. The standard deviations are
s1 = .12 and s2 = .22
Thus,
V(Rp) = w21s21 + w22s22 + 2w1w2rs1s2
= (.252)(.122) + (.752)(.222) + 2(.25)(.75) r (.12)(.22)
= .0281 + .0099r
When ␳ ⫽ 1
V(Rp) = .0281 + .0099(1) = .0380
Standard deviation = 2V(Rp) = 2.0380 = .1949
When ␳ ⫽ .5
V(Rp) = .0281 + .0099(.5) = .0331
Standard deviation = 2V(Rp) = 2.0331 = .1819
When ␳ ⫽ 0
V(Rp) = .0281 + .0099(0) = .0281
Standard deviation = 2V(Rp) = 2.0281 = .1676
Notice that the variance and standard deviation of the portfolio returns decrease as
the coefficient of correlation decreases.
Portfolio Diversification in Practice
The formulas introduced in this section require that we know the expected values,
variances, and covariance (or coefficient of correlation) of the investments we’re
interested in. The question arises, how do we determine these parameters?
(Incidentally, this question is rarely addressed in finance textbooks!) The most common procedure is to estimate the parameters from historical data, using sample
statistics.
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Portfolios with More Than Two Stocks
We can extend the formulas that describe the mean and variance of the returns of a
portfolio of two stocks to a portfolio of any number of stocks.
Mean and Variance of a Portfolio of k Stocks
k
E(Rp) = a wiE(Ri)
i=1
k
k
k
V(Rp) = a w2i s2i + 2 a a wiwjCOV(Ri, Rj)
i=1
i=1 j=i+1
Where Ri is the return of the ith stock, wi is the proportion of the portfolio
invested in stock i, and k is the number of stocks in the portfolio.
Investing to Maximize Returns and
Minimize Risk: Solution
Because of the large number of calculations, we will solve this problem using only Excel. From
the file, we compute the means of each stock’s returns.
Excel Means
1
A
0.00881
B
0.00562
C
0.01253
© Terry Vine/Blend Images/
Jupiterimages
When k is greater than 2, the calculations can be tedious and time consuming. For
example, when k ⫽ 3, we need to know the values of the three weights, three expected
values, three variances, and three covariances. When k ⫽ 4, there are four expected values, four variances, and six covariances. [The number of covariances required in general
is k(k ⫺ 1)/2.] To assist you, we have created an Excel worksheet to perform the computations when k ⫽ 2, 3, or 4. To demonstrate, we’ll return to the problem described in
this chapter’s introduction.
D
0.02834
Next we compute the variance–covariance matrix. (The commands are the same as those described
in Chapter 4: Simply include all the columns of the returns of the investments you wish to include in
the portfolio.)
Excel Variance-Covariance Matrix
A
1
2
3
4
5
B
C
Coca Cola
Disney
Coca Cola 0.00235
Disney
0.00141
0.00434
Barrick
0.00184 –0.00058
Amazon
0.00167
0.00182
D
Barrick
E
Amazon
0.01174
–0.00170
0.02020
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
241
Notice that the variances of the returns are listed on the diagonal. Thus, for example, the variance of the 60 monthly returns
of Barrick Gold is .01174. The covariances appear below the diagonal. The covariance between the returns of Coca Cola and
Disney is .00141.
The means and the variance–covariance matrix are copied to the spreadsheet using the commands described here.
The weights are typed producing the accompanying output.
Excel Worksheet: Portfolio Diversification—Plan 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Portfolio of 4 Stocks
B
C
Variance-Covariance Matrix Coca Cola
Disney
Barrick
Amazon
D
Coca Cola Disney
0.00235
0.00141
0.00434
0.00184 –0.00058
0.00167
0.00182
E
F
Barrick
Amazon
0.01174
–0.00170
0.02020
Expected Returns
0.00881
0.00562
0.01253
0.02834
Weights
0.25000
0.25000
0.25000
0.25000
Portfolio Return
Expected Value
Variance
Standard Deviation
0.01382
0.00297
0.05452
The expected return on the portfolio is .01382, and the variance is .00297.
INSTRUCTIONS
1. Open the file containing the returns. In this example, open file Ch7:\ Xm07-00
2. Compute the means of the columns containing the returns of the stocks in the portfolio.
3. Using the commands described in Chapter 4 (page 137) compute the variance–covariance matrix.
4. Open the Portfolio Diversification workbook. Use the tab to select the 4 Stocks worksheet. Do not change any cells
that appear in bold print. Do not save any worksheets.
5. Copy the means into cells C8 to F8. (Use Copy, Paste Special with Values and number formats.)
6. Copy the variance–covariance matrix (including row and column labels) into columns B, C, D, E, and F.
7. Type the weights into cells C10 to F10
The mean, variance, and standard deviation of the portfolio will be printed. Use similar commands for 2 stock and 3 stock
portfolios.
The results for plan 2 are
12
13
14
15
A
Portfolio Return
Expected Value
Variance
Standard Deviation
B
0.01710
0.00460
0.06783
Plan 3
12
13
14
15
A
Portfolio Return
Expected Value
Variance
Standard Deviation
B
0.01028
0.00256
0.05059
Plan 3 has the smallest expected value and the smallest variance. Plan 2 has the largest expected value and the largest
variance. Plan 1’s expected value and variance are in the middle. If the investor is like most investors, she would select
Plan 3 because of its lower risk. Other, more daring investors may choose plan 2 to take advantage of its higher expected
value.
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In this example, we showed how to compute the expected return, variance, and standard
deviation from a sample of returns on the investments for any combination of weights.
(We illustrated the process with three sets of weights.) It is possible to determine the
“optimal” weights that minimize risk for a given expected value or maximize expected
return for a given standard deviation. This is an extremely important function of financial analysts and investment advisors. Solutions can be determined using a managementscience technique called linear programming, a subject taught by most schools of
business and faculties of management.
EXERCISES
7.61 Describe what happens to the expected value and
standard deviation of the portfolio returns when the
coefficient of correlation decreases.
7.62 A portfolio is composed of two stocks. The propor-
tion of each stock, their expected values, and standard deviations are listed next.
Stock
Proportion of portfolio
Mean
Standard deviation
1
.30
.12
.02
2
.70
.25
.15
For each of the following coefficients of correlation
calculate the expected value and standard deviation
of the portfolio.
a. ␳ ⫽ .5
b. ␳ ⫽ .2
c. ␳ ⫽ 0
7.63 An investor is given the following information about
the returns on two stocks.
Stock
Mean
Standard deviation
1
.09
.15
2
.13
.21
a. If he is most interested in maximizing his returns,
which stock should he choose?
b. If he is most interested in minimizing his risk,
which stock should he choose?
7.64 Refer to Exercise 7.63. Compute the expected value
and standard deviation of the portfolio composed of
60% stock 1 and 40% stock 2. The coefficient of
correlation is .4.
7.65 Refer to Exercise 7.63. Compute the expected value
and standard deviation of the portfolio composed of
30% stock 1 and 70% stock 2.
The following exercises require the use of a computer.
The monthly returns for the following stocks on the
New York Stock Exchange were recorded.
AT&T, Aetna, Cigna, Coca-Cola, Disney, Ford, and McDonald’s
The next seven exercises are based on this set of data.
7.66 a. Calculate the mean and variance of the monthly
return for each stock.
b. Determine the variance–covariance matrix.
7.67 Select the two stocks with the largest means and
construct a portfolio consisting of equal amounts of
both. Determine the expected value and standard
deviation of the portfolio.
7.68 Select the two stocks with the smallest variances and
construct a portfolio consisting of equal amounts of
both. Determine the expected value and standard
deviation of the portfolio.
7.69 Describe the results of Exercises 7.66 to 7.68.
7.70 An investor wants to develop a portfolio composed
of shares of AT&T, Coca-Cola, Ford, and Disney.
Calculate the expected value and standard deviation
of the returns for a portfolio with equal proportions
of all three stocks.
7.71 Suppose you want a portfolio composed of AT&T,
Cigna, Disney, and Ford. Find the expected value
and standard deviation of the returns for the following portfolio.
AT&T
Cigna
Disney
Ford
30%
20%
40%
10%
7.72 Repeat Exercise 7.71 using the following proportions.
Compare your results with those of Exercise 7.71.
AT&T
Cigna
Disney
Ford
30%
10%
40%
20%
Xr07-66
The following seven exercises are directed at Canadian students.
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243
Xr07-73 The monthly returns for the following stocks on the
Toronto Stock Exchange were recorded:
Xr07-80
Barrick Gold, Bell Canada Enterprises (BCE),Bank of Montreal
(BMO), Enbridge, Fortis, Methanex, Research in Motion, Telus,
and Trans Canada Pipeline
Amazon, Amgen, Apple, Cisco Systems, Google, Intel, Microsoft,
Oracle, and Research in Motion
The next seven exercises are based on this set of data.
7.73 a. Calculate the mean and variance of the monthly
return for each stock.
b. Determine the correlation matrix.
7.74 Select the two stocks with the largest means and
construct a portfolio consisting of equal amounts of
both. Determine the expected value and standard
deviation of the portfolio.
7.75 Select the two stocks with the smallest variances and
construct a portfolio consisting of equal amounts of
both. Determine the expected value and standard
deviation of the portfolio.
7.76 Describe the results of Exercises 7.73 to 7.75.
7.77 An investor wants to develop a portfolio composed
of shares of Bank of Montreal, Enbridge, and Fortis.
Calculate the expected value and standard deviation
of the returns for a portfolio with the following proportions.
Bank of Montreal
Enbridge
Fortis
20%
30%
50%
7.78 Suppose you want a portfolio composed of Barrick
Gold, Bell Canada Enterprises, Telus, and TransCanada Ltd. Find the expected value and standard
deviation of the returns for the following portfolio.
Barrick Gold
Bell Canada Enterprises
Telus
TransCanada
50%
25%
15%
10%
7.79 Repeat Exercise 7.78 using the following proportions.
Compare your results with those of Exercise 7.78.
Barrick Gold
Bell Canada Enterprises
Telus
TransCanada
20%
40%
20%
20%
The monthly returns for the following stocks on the
NASDAQ Stock Exchange were recorded:
The next four exercises are based on this set of data.
7.80 a. Calculate the mean and variance of the monthly
return for each stock.
b. Determine which four stocks you would include
in your portfolio if you wanted a large expected
value.
c. Determine which four stocks you would include
in your portfolio if you wanted a small variance.
7.81 Suppose you want a portfolio composed of Cisco
Systems, Intel, Microsoft, and Research in Motion.
Find the expected value and standard deviation of
the returns for the following portfolio.
Cisco Systems
Intel
Microsoft
Research in Motion
30%
15%
25%
30%
7.82 An investor wants to acquire a portfolio composed
of Cisco Systems, Intel, Microsoft, and Research in
Motion. Moreover, he wants the expected value to
be at least 1%. Try several sets of proportions
(remember, they must add to 1.0) to see if you can
find the portfolio with the smallest variance.
7.83 Refer to Exercise 7.81.
a. Compute the expected value and variance of the
portfolio described next.
Cisco Systems
Intel
Microsoft
Research in Motion
26.59%
2.49%
54.74%
16.19%
b. Can you do better? In other words, can you find
a portfolio whose expected value is greater than
or equal to 1% and whose variance is less than
the one you calculated in part (a)? (Hint: Don’t
spend too much time at this. You won’t be able to
do better.)
c. If you want to learn how we produced the portfolio above, take a course that teaches linear and
nonlinear programming.
7. 4 B I N O M I A L D I S T R I B U T I O N
Now that we’ve introduced probability distributions in general, we need to introduce several specific probability distributions. In this section, we present the binomial distribution.
The binomial distribution is the result of a binomial experiment, which has the
following properties.
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Binomial Experiment
1. The binomial experiment consists of a fixed number of trials. We
represent the number of trials by n.
2. Each trial has two possible outcomes. We label one outcome a success,
and the other a failure.
3. The probability of success is p. The probability of failure is 1 ⫺ p.
4. The trials are independent, which means that the outcome of one trial
does not affect the outcomes of any other trials.
If properties 2, 3, and 4 are satisfied, we say that each trial is a Bernoulli process.
Adding property 1 yields the binomial experiment. The random variable of a binomial
experiment is defined as the number of successes in the n trials. It is called the binomial
random variable. Here are several examples of binomial experiments.
1.
Flip a coin 10 times. The two outcomes per trial are heads and tails. The terms
success and failure are arbitrary. We can label either outcome success. However,
generally, we call success anything we’re looking for. For example, if we were betting on heads, we would label heads a success. If the coin is fair, the probability of
heads is 50%. Thus, p ⫽ .5. Finally, we can see that the trials are independent
because the outcome of one coin flip cannot possibly affect the outcomes of other
flips.
2.
Draw five cards out of a shuffled deck. We can label as success whatever card we
seek. For example, if we wish to know the probability of receiving five clubs, a club
is labeled a success. On the first draw, the probability of a club is 13/52 ⫽ .25.
However, if we draw a second card without replacing the first card and shuffling,
the trials are not independent. To see why, suppose that the first draw is a club. If
we draw again without replacement the probability of drawing a second club is
12/51, which is not .25. In this experiment, the trials are not independent.* Hence,
this is not a binomial experiment. However, if we replace the card and shuffle
before drawing again, the experiment is binomial. Note that in most card games,
we do not replace the card, and as a result the experiment is not binomial.
3.
A political survey asks 1,500 voters who they intend to vote for in an approaching
election. In most elections in the United States, there are only two candidates, the
Republican and Democratic nominees. Thus, we have two outcomes per trial.
The trials are independent because the choice of one voter does not affect the
choice of other voters. In Canada, and in other countries with parliamentary systems of government, there are usually several candidates in the race. However, we
can label a vote for our favored candidate (or the party that is paying us to do the
survey) a success and all the others are failures.
As you will discover, the third example is a very common application of statistical inference. The actual value of p is unknown, and the job of the statistics practitioner is to
estimate its value. By understanding the probability distribution that uses p, we will be
able to develop the statistical tools to estimate p.
*The hypergeometric distribution described in Keller’s website Appendix of the same name is used to
calculate probabilities in such cases.
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245
Binomial Random Variable
The binomial random variable is the number of successes in the experiment’s n trials. It
can take on values 0, 1, 2, . . . , n. Thus, the random variable is discrete. To proceed, we
must be capable of calculating the probability associated with each value.
Using a probability tree, we draw a series of branches as depicted in Figure 7.2.
The stages represent the outcomes for each of the n trials. At each stage, there are two
branches representing success and failure. To calculate the probability that there are
X successes in n trials, we note that for each success in the sequence we must multiply
by p. And if there are X successes, there must be n ⫺ X failures. For each failure in the
sequence, we multiply by 1 ⫺ p. Thus, the probability for each sequence of branches
that represent x successes and n ⫺ x failures has probability
px(1 - p)n - x
FIGURE
7.2
Trial 1
Probability Tree for a Binomial Experiment
Trial 2
Trial 3
S
. . .
Trial n
S
S
F
F
S
F
S
F
S
F
F
S
.
.
.
S
F
S
F
F
S
S
F
F
There are a number of branches that yield x successes and n ⫺ x failures. For example, there are two ways to produce exactly one success and one failure in two trials:
SF and FS. To count the number of branch sequences that produce x successes and
n ⫺ x failures, we use the combinatorial formula
Cnx =
n!
x!(n - x)!
where n! ⫽ n(n ⫺ 1) (n ⫺ 2) . . . (2)(1). For example, 3! ⫽ 3(2)(1) ⫽ 6. Incidentally,
although it may not appear to be logical 0! ⫽ 1.
Pulling together the two components of the probability distribution yields the
following.
Binomial Probability Distribution
The probability of x successes in a binomial experiment with n trials and
probability of success ⫽ p is
P(x) =
n!
p x(1 - p)n - x for x = 0, 1, 2, Á , n
x!(n - x)!
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CHAPTER 7
EXAMPLE
7.9
Pat Statsdud and the Statistics Quiz
Pat Statsdud is a student taking a statistics course. Unfortunately, Pat is not a good student. Pat does not read the textbook before class, does not do homework, and regularly
misses class. Pat intends to rely on luck to pass the next quiz. The quiz consists of
10 multiple-choice questions. Each question has five possible answers, only one of
which is correct. Pat plans to guess the answer to each question.
a. What is the probability that Pat gets no answers correct?
b. What is the probability that Pat gets two answers correct?
SOLUTION
The experiment consists of 10 identical trials, each with two possible outcomes and where
success is defined as a correct answer. Because Pat intends to guess, the probability of
success is 1/5 or .2. Finally, the trials are independent because the outcome of any of the
questions does not affect the outcomes of any other questions. These four properties tell
us that the experiment is binomial with n ⫽ 10 and p ⫽ .2.
a. From
P(x) =
n!
px(1 - p)n - x
x!(n - x)!
we produce the probability of no successes by letting n ⫽ 10, p ⫽ .2, and x ⫽ 0. Hence,
P(0) =
10!
(.2)0(1 - .2)10 - 0
0!(10 - 0)!
10!
, which is 1. This is the number of ways
0!10!
to get 0 correct and 10 incorrect. Obviously, there is only one way to produce X ⫽ 0.
And because (.2)0 ⫽ 1,
The combinatorial part of the formula is
P(X = 0) = 1(1)(.8)10 = .1074
b. The probability of two correct answers is computed similarly by substituting
n ⫽ 10, p ⫽ .2, and x ⫽ 2:
P(x) =
n!
px(1 - p)n - x
x!(n - x)!
P(0) =
10!
(.2)2(1 - .2)10 - 2
2!(10 - 2)!
(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)
=
(2)(1)(8)(7)(6)(5)(4)(3)(2)(1)
(.04)(.1678)
= 45(.006712)
= .3020
In this calculation, we discovered that there are 45 ways to get exactly two correct and
eight incorrect answers, and that each such outcome has probability .006712.
Multiplying the two numbers produces a probability of .3020.
Cumulative Probability
The formula of the binomial distribution allows us to determine the probability that X
equals individual values. In Example 7.9, the values of interest were 0 and 2. There are
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
247
many circumstances where we wish to find the probability that a random variable is less
than or equal to a value; that is, we want to determine P(X … x), where x is that value.
Such a probability is called a cumulative probability.
EXAMPLE
7. 1 0
Will Pat Fail the Quiz?
Find the probability that Pat fails the quiz. A mark is considered a failure if it is less
than 50%.
SOLUTION
In this quiz, a mark of less than 5 is a failure. Because the marks must be integers, a
mark of 4 or less is a failure. We wish to determine P(X … 4). So,
P(X … 4) = P(0) + P(1) + P(2) + P(3) + P(4)
From Example 7.9, we know P(0) ⫽ .1074 and P(2) ⫽ .3020. Using the binomial formula, we find P(1) ⫽ .2684 , P(3) ⫽ .2013, and P(4) ⫽ .0881. Thus
P(X … 4) = .1074 + .2684 + .3020 + .2013 + .0881 = .9672
There is a 96.72% probability that Pat will fail the quiz by guessing the answer for each
question.
Binomial Table
There is another way to determine binomial probabilities. Table 1 in Appendix B provides
cumulative binomial probabilities for selected values of n and p. We can use this table to
answer the question in Example 7.10, where we need P(X … 4). Refer to Table 1, find
n ⫽ 10, and in that table find p ⫽ .20. The values in that column are P(X … x) for
x ⫽ 0, 1, 2, . . . , 10, which are shown in Table 7.2.
TABLE
7. 2
Cumulative Binomial Probabilities with n ⫽ 10 and p ⫽ .2
x
P (X … x)
0
.1074
1
.3758
2
.6778
3
.8791
4
.9672
5
.9936
6
.9991
7
.9999
8
1.000
9
1.000
10
1.000
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CHAPTER 7
The first cumulative probability is P(X … 0), which is P(0) ⫽ .1074. The probability we
need for Example 7.10 is P(X … 4) = .9672, which is the same value we obtained
manually.
We can use the table and the complement rule to determine probabilities of the type
P(X Ú x). For example, to find the probability that Pat will pass the quiz, we note that
P(X … 4) + P(X Ú 5) = 1
Thus,
P(X Ú 5) = 1 - P(X … 4) = 1 - .9672 = .0328
Using Table 1 to Find the Binomial Probability P(X Ú x)
P(X Ú x) = 1 - P(X … 3x - 14)
The table is also useful in determining the probability of an individual value of X. For
example, to find the probability that Pat will get exactly two right answers we note that
P(X … 2) = P(0) + P(1) + P(2)
and
P(X … 1) = P(0) + P(1)
The difference between these two cumulative probabilities is p(2). Thus,
P(2) = P(X … 2) - P(X … 1) = .6778 - .3758 = .3020
Using Table 1 to Find the Binomial Probability P(X ⫽ x)
P(x) = P(X … x) - P(X … 3x - 14)
Using the Computer
EXCEL
INSTRUCTIONS
Type the following into any empty cell:
=BINOMDIST(3x4, 3n4, 3p4, 3True4 or 3False4)
Typing “True” calculates a cumulative probability and typing “False” computes the probability of an individual value of X. For Example 7.9(a), type
=BINOMDIST(0, 10, .2, False)
For Example 7.10, enter
=BINOMDIST(4, 10, .2, True)
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249
MINITAB
INSTRUCTIONS
This is the first of seven probability distributions for which we provide instructions. All
work in the same way. Click Calc, Probability Distributions, and the specific distribution whose probability you wish to compute. In this case, select Binomial . . . . Check
either Probability or Cumulative probability. If you wish to make a probability statement about one value of x, specify Input constant and type the value of x.
If you wish to make probability statements about several values of x from the same
binomial distribution, type the values of x into a column before checking Calc. Choose
Input column and type the name of the column. Finally, enter the components of the distribution. For the binomial, enter the Number of trials n and the Event Probability p.
For the other six distributions, we list the distribution (here it is Binomial) and the
components only (for this distribution it is n and p).
Mean and Variance of a Binomial Distribution
Statisticians have developed general formulas for the mean, variance, and standard deviation of a binomial random variable. They are
m = np
s2 = np(1 - p)
s = 2np(1 - p)
EXAMPLE
Pat Statsdud Has Been Cloned!
7. 1 1
Suppose that a professor has a class full of students like Pat (a nightmare!). What is the
mean mark? What is the standard deviation?
SOLUTION
The mean mark for a class of Pat Statsduds is
m = np = 10(.2) = 2
The standard deviation is
s = 2np(1 - p) = 210(.2)(1 - .2) = 1.26
EXERCISES
7.84 Given a binomial random variable with n ⫽ 10
and p ⫽ .3, use the formula to find the following
probabilities.
a. P(X ⫽ 3)
b. P(X ⫽ 5)
c. P(X ⫽ 8)
7.85 Repeat Exercise 7.84 using Table 1 in Appendix B.
7.86 Repeat Exercise 7.84 using Excel or Minitab.
7.87 Given a binomial random variable with n ⫽ 6
and p ⫽ .2, use the formula to find the following
probabilities.
a. P(X ⫽ 2)
b. P(X ⫽ 3)
c. P(X ⫽ 5)
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7.88 Repeat Exercise 7.87 using Table 1 in Appendix B.
7.89 Repeat Exercise 7.87 using Excel or Minitab.
7.90 Suppose X is a binomial random variable with n ⫽ 25
and p ⫽ .7. Use Table 1 to find the following.
a. P(X ⫽ 18)
b. P(X ⫽ 15)
c. P(X … 20)
d. P(X Ú 16)
7.91 Repeat Exercise 7.90 using Excel or Minitab.
7.92 A sign on the gas pumps of a chain of gasoline sta-
tions encourages customers to have their oil checked
with the claim that one out of four cars needs to
have oil added. If this is true, what is the probability
of the following events?
a. One out of the next four cars needs oil
b. Two out of the next eight cars need oil
c. Three out of the next 12 cars need oil
7.93 The leading brand of dishwasher detergent has a
30% market share. A sample of 25 dishwasher detergent customers was taken. What is the probability
that 10 or fewer customers chose the leading brand?
7.94 A certain type of tomato seed germinates 90% of the
time. A backyard farmer planted 25 seeds.
a. What is the probability that exactly 20 germinate?
b. What is the probability that 20 or more germinate?
c. What is the probability that 24 or fewer germinate?
d. What is the expected number of seeds that
germinate?
7.95 According to the American Academy of Cosmetic
Dentistry, 75% of adults believe that an unattractive
smile hurts career success. Suppose that 25 adults
are randomly selected. What is the probability that
15 or more of them would agree with the claim?
7.96 A student majoring in accounting is trying to decide
on the number of firms to which he should apply.
Given his work experience and grades, he can expect
to receive a job offer from 70% of the firms to which
he applies. The student decides to apply to only four
firms. What is the probability that he receives no job
offers?
7.97 In the United States, voters who are neither Dem-
ocrat nor Republican are called Independents. It is
believed that 10% of all voters are Independents.
A survey asked 25 people to identify themselves as
Democrat, Republican, or Independent.
a. What is the probability that none of the people
are Independent?
b. What is the probability that fewer than five people are Independent?
c. What is the probability that more than two people are Independent?
7.98 Most dial-up Internet service providers (ISPs)
attempt to provide a large enough service so that
customers seldom encounter a busy signal. Suppose
that the customers of one ISP encounter busy signals 8% of the time. During the week, a customer of
this ISP called 25 times. What is the probability that
she did not encounter any busy signals?
7.99 Major software manufacturers offer a help line that
allows customers to call and receive assistance in
solving their problems. However, because of the volume of calls, customers frequently are put on hold.
One software manufacturer claims that only 20% of
callers are put on hold. Suppose that 100 customers
call. What is the probability that more than 25 of
them are put on hold?
7.100 A statistics practitioner working for major league
baseball determined the probability that the hitter
will be out on ground balls is .75. In a game where
there are 20 ground balls, find the probability that
all of them were outs.
The following exercises are best solved with a computer.
7.101 The probability of winning a game of craps (a dice-
throwing game played in casinos) is 244/495.
a. What is the probability of winning 5 or more
times in 10 games?
b. What is the expected number of wins in 100
games?
7.102 In the game of blackjack as played in casinos in Las
Vegas, Atlantic City, and Niagara Falls, as well as in
many other cities, the dealer has the advantage. Most
players do not play very well. As a result, the probability that the average player wins a hand is about 45%.
Find the probability that an average player wins.
a. Twice in 5 hands
b. Ten or more times in 25 hands
7.103 Several books teach blackjack players the “basic
strategy,” which increases the probability of winning
any hand to 50%. Repeat Exercise 7.102, assuming
the player plays the basic strategy.
7.104 The best way of winning at blackjack is to “case the
deck,” which involves counting 10s, non-10s, and
aces. For card counters, the probability of winning a
hand may increase to 52%. Repeat Exercise 7.102
for a card counter.
7.105 In the game of roulette, a steel ball is rolled onto a
wheel that contains 18 red, 18 black, and 2 green
slots. If the ball is rolled 25 times, find the probabilities of the following events.
a. The ball falls into the green slots two or more
times.
b. The ball does not fall into any green slots.
c. The ball falls into black slots 15 or more times.
d. The ball falls into red slots 10 or fewer times.
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7.106 According to a Gallup Poll conducted March 5–7,
2001, 52% of American adults think that protecting
the environment should be given priority over developing U.S. energy supplies. Thirty-six percent think
that developing energy supplies is more important,
and 6% believe the two are equally important. The
rest had no opinion. Suppose that a sample of
100 American adults is quizzed on the subject. What
is the probability of the following events?
a. Fifty or more think that protecting the environment should be given priority.
b. Thirty or fewer think that developing energy
supplies is more important.
c. Five or fewer have no opinion.
7.107 In a Bon Appetit poll, 38% of people said that choco-
251
of 20 people was asked to name their favorite flavor
of ice cream. What is the probability that half or
more of them prefer chocolate?
7.108 The statistics practitioner in Exercise 7.100 also deter-
mined that if a batter hits a line drive, the probability
of an out is .23. Determine the following probabilities.
a. In a game with 10 line drives, at least 5 are outs.
b. In a game with 25 line drives, there are 5 outs or
less.
7.109 According to the last census, 45% of working
women held full-time jobs in 2002. If a random
sample of 50 working women is drawn, what is
the probability that 19 or more hold full-time
jobs?
late was their favorite flavor of ice cream. A sample
7. 5
POISSON DISTRIBUTION
Another useful discrete probability distribution is the Poisson distribution, named
after its French creator. Like the binomial random variable, the Poisson random variable is the number of occurrences of events, which we’ll continue to call successes. The
difference between the two random variables is that a binomial random variable is the
number of successes in a set number of trials, whereas a Poisson random variable is
the number of successes in an interval of time or specific region of space. Here are several examples of Poisson random variables.
1.
2.
3.
The number of cars arriving at a service station in 1 hour. (The interval of time is
1 hour.)
The number of flaws in a bolt of cloth. (The specific region is a bolt of cloth.)
The number of accidents in 1 day on a particular stretch of highway. (The interval is defined by both time, 1 day, and space, the particular stretch of highway.)
The Poisson experiment is described in the box.
Poisson Experiment
A Poisson experiment is characterized by the following properties:
1. The number of successes that occur in any interval is independent of the
number of successes that occur in any other interval.
2. The probability of a success in an interval is the same for all equal-size
intervals.
3. The probability of a success in an interval is proportional to the size of
the interval.
4. The probability of more than one success in an interval approaches 0 as
the interval becomes smaller.
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CHAPTER 7
Poisson Random Variable
The Poisson random variable is the number of successes that occur in a
period of time or an interval of space in a Poisson experiment.
There are several ways to derive the probability distribution of a Poisson random
variable. However, all are beyond the mathematical level of this book. We simply provide the formula and illustrate how it is used.
Poisson Probability Distribution
The probability that a Poisson random variable assumes a value of x in a
specific interval is
P(x) =
e - mmx
for x = 0,1,2, Á
x!
where ␮ is the mean number of successes in the interval or region and e is
the base of the natural logarithm (approximately 2.71828). Incidentally, the
variance of a Poisson random variable is equal to its mean; that is, ␴2 ⫽ ␮.
EXAMPLE
7. 1 2
Probability of the Number of Typographical Errors in
Textbooks
A statistics instructor has observed that the number of typographical errors in new
editions of textbooks varies considerably from book to book. After some analysis, he
concludes that the number of errors is Poisson distributed with a mean of 1.5 per
100 pages. The instructor randomly selects 100 pages of a new book. What is the probability that there are no typographical errors?
SOLUTION
We want to determine the probability that a Poisson random variable with a mean of
1.5 is equal to 0. Using the formula
P(x) =
e - mmx
x!
and substituting x ⫽ 0 and ␮ ⫽ 1.5, we get
P(0) =
(2.71828) - 1.5(1)
e - 1.51.50
=
= .2231
0!
1
The probability that in the 100 pages selected there are no errors is .2231.
Notice that in Example 7.12 we wanted to find the probability of 0 typographical
errors in 100 pages given a mean of 1.5 typos in 100 pages. The next example illustrates how we calculate the probability of events where the intervals or regions do not
match.
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
EXAMPLE
7. 1 3
253
Probability of the Number of Typographical Errors
in 400 Pages
Refer to Example 7.12. Suppose that the instructor has just received a copy of a new
statistics book. He notices that there are 400 pages.
a. What is the probability that there are no typos?
b. What is the probability that there are five or fewer typos?
SOLUTION
The specific region that we’re interested in is 400 pages. To calculate Poisson probabilities associated with this region, we must determine the mean number of typos per
400 pages. Because the mean is specified as 1.5 per 100 pages, we multiply this figure by
4 to convert to 400 pages. Thus, ␮ ⫽ 6 typos per 400 pages.
a. The probability of no typos is
P(0) =
(2.71828) - 6(1)
e - 660
=
= .002479
0!
1
b. We want to determine the probability that a Poisson random variable with a
mean of 6 is 5 or less; that is, we want to calculate
P(X … 5) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5)
To produce this probability, we need to compute the six probabilities in the summation.
P(0) = .002479
P(1) =
P(2) =
P(3) =
P(4) =
P(5) =
e - mmx
x!
e - mmx
x!
e - mmx
x!
e - mmx
x!
e - mmx
x!
=
=
=
=
=
(2.71828) - 6(6)
e - 661
=
= .01487
1!
1
(2.71828) - 6(36)
e - 662
=
= .04462
2!
2
(2.71828) - 6(216)
e - 663
= .08924
=
3!
6
(2.71828) - 6(1296)
e - 664
=
= .1339
4!
24
(2.71828) - 6(7776)
e - 665
=
= .1606
5!
120
Thus,
P(X … 5) = .002479 + .01487 + .04462 + .08924 + .1339 + .1606
= .4457
The probability of observing 5 or fewer typos in this book is .4457.
Poisson Table
As was the case with the binomial distribution, a table is available that makes it easier to
compute Poisson probabilities of individual values of x as well as cumulative and related
probabilities.
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CHAPTER 7
Table 2 in Appendix B provides cumulative Poisson probabilities for selected values
of ␮. This table makes it easy to find cumulative probabilities like those in Example 7.13,
part (b), where we found P(X … 5).
To do so, find ␮ ⫽ 6 in Table 2. The values in that column are P(X … x) for
x ⫽ 0, 1, 2, . . . , 18 which are shown in Table 7.3.
TABLE
7. 3
Cumulative Poisson Probabilities for ␮ ⫽ 6
x
P (X … x)
0
.0025
1
.0174
2
.0620
3
.1512
4
.2851
5
.4457
6
.6063
7
.7440
8
.8472
9
.9161
10
.9574
11
.9799
12
.9912
13
.9964
14
.9986
15
.9995
16
.9998
17
.9999
18
1.0000
Theoretically, a Poisson random variable has no upper limit. The table provides
cumulative probabilities until the sum is 1.0000 (using four decimal places).
The first cumulative probability is P(X … 0), which is P(0) ⫽ .0025. The probability we need for Example 7.13, part (b), is P(X … 5) = .4457, which is the same value we
obtained manually.
Like Table 1 for binomial probabilities, Table 2 can be used to determine probabilities of the type P(X Ú x). For example, to find the probability that in Example 7.13
there are 6 or more typos, we note that P(X … 5) + P(X Ú 6) = 1. Thus,
P(X Ú 6) = 1 - P(X … 5) = 1 - .4457 = .5543
Using Table 2 to Find the Poisson Probability P(X Ú x)
P(X Ú x) = 1 - P(X … 3x - 14)
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
255
We can also use the table to determine the probability of one individual value of X.
For example, to find the probability that the book contains exactly 10 typos, we note
that
P(X … 10) = P(0) + P(1) + Á + P(9) + P(10)
and
P(X … 9) = P(0) + P(1) + Á + P(9)
The difference between these two cumulative probabilities is P(10). Thus,
P(10) = P(X … 10) - P(X … 9) = .9574 - .9161 = .0413
Using Table 2 to Find the Poisson Probability P(X ⫽ x)
P(x) = P(X … x) - P(X … 3x - 14)
Using the Computer
EXCEL
INSTRUCTIONS
Type the following into any empty cell:
= POISSON(3x4, 3m4, 3True] or [False4)
We calculate the probability in Example 7.12 by typing
= POISSON(0, 1.5, False)
For Example 7.13, we type
= POISSON(5, 6, True)
MINITAB
INSTRUCTIONS
Click Calc, Probability Distributions, and Poisson . . . and type the mean.
EXERCISES
7.110 Given a Poisson random variable with ␮ ⫽ 2, use
the formula to find the following probabilities.
a. P(X ⫽ 0)
b. P(X ⫽ 3)
c. P(X ⫽ 5)
7.111 Given that X is a Poisson random variable with
␮ ⫽ .5, use the formula to determine the following
probabilities.
a. P(X ⫽ 0)
b. P(X ⫽ 1)
c. P(X ⫽ 2)
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CHAPTER 7
7.112 The number of accidents that occur at a busy inter-
homes and businesses. A report to the Toronto city
council stated that there are on average 30 water line
breaks per 100 kilometers per year in the city of
Toronto. Outside of Toronto, the average number of
breaks is 15 per 100 kilometers per year.
a. Find the probability that in a stretch of 100 kilometers in Toronto there are 35 or more breaks
next year.
b. Find the probability that there are 12 or fewer
breaks in a stretch of 100 kilometers outside of
Toronto next year.
section is Poisson distributed with a mean of 3.5
per week. Find the probability of the following
events.
a. No accidents in one week
b. Five or more accidents in one week
c. One accident today
7.113 Snowfalls occur randomly and independently over
the course of winter in a Minnesota city. The average is one snowfall every 3 days.
a. What is the probability of five snowfalls in
2 weeks?
b. Find the probability of a snowfall today.
7.117 The number of bank robberies that occur in a large
North American city is Poisson distributed with a
mean of 1.8 per day. Find the probabilities of the following events.
a. Three or more bank robberies in a day
b. Between 10 and 15 (inclusive) robberies during a
5-day period
7.114 The number of students who seek assistance with
their statistics assignments is Poisson distributed
with a mean of two per day.
a. What is the probability that no students seek
assistance tomorrow?
b. Find the probability that 10 students seek assistance in a week.
7.118 Flaws in a carpet tend to occur randomly and inde-
pendently at a rate of one every 200 square feet.
What is the probability that a carpet that is 8 feet by
10 feet contains no flaws?
7.115 Hits on a personal website occur quite infrequently.
They occur randomly and independently with an
average of five per week.
a. Find the probability that the site gets 10 or more
hits in a week.
b. Determine the probability that the site gets 20 or
more hits in 2 weeks.
7.119 Complaints about an Internet brokerage firm occur
7.116 In older cities across North America, infrastructure
is deteriorating, including water lines that supply
at a rate of five per day. The number of complaints
appears to be Poisson distributed.
a. Find the probability that the firm receives 10 or
more complaints in a day.
b. Find the probability that the firm receives 25 or
more complaints in a 5-day period.
APPLICATIONS in OPERATIONS MANAGEMENT
© Lester Lefkowitz/Getty Images
Waiting Lines
Everyone is familiar with waiting lines. We wait in line at banks, groceries, and
fast-food restaurants. There are also waiting lines in firms where trucks wait to
load and unload and on assembly lines where stations wait for new parts.
Management scientists have developed mathematical models that allow managers to determine the operating characteristics of waiting lines. Some of the
operating characteristics are
The probability that there are no units in the system
The average number of units in the waiting line
The average time a unit spends in the waiting line
The probability that an arriving unit must wait for service
The Poisson probability distribution is used extensively in waiting-line (also called queuing) models. Many models assume that the arrival of units for service is Poisson distributed with a specific
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
257
value of ␮. In the next chapter, we will discuss the operating characteristics of waiting lines.
Exercises 7.120–7.122 require the calculation of the probability of a number of arrivals.
7.120 The number of trucks crossing at the Ambassador Bridge connecting Detroit,
Michigan, and Windsor, Ontario, is Poisson distributed with a mean of 1.5 per
minute.
a. What is the probability that in any 1-minute time span two or more trucks will
cross the bridge?
b. What is the probability that fewer than four trucks will cross the bridge over
the next 4 minutes?
7.121 Cars arriving for gasoline at a particular gas station follow a Poisson distribution
with a mean of 5 per hour.
a. Determine the probability that over the next hour only one car will arrive.
b. Compute the probability that in the next 3 hours more than 20 cars will arrive.
7.122 The number of users of an automatic banking machine is Poisson distributed. The
mean number of users per 5-minute interval is 1.5. Find the probability of the
following events.
a. No users in the next 5 minutes
b. Five or fewer users in the next 15 minutes
c. Three or more users in the next 10 minutes
CHAPTER SUM M ARY
There are two types of random variables. A discrete random variable is one whose values are countable. A continuous random variable can assume an uncountable
number of values. In this chapter, we discussed discrete
random variables and their probability distributions. We
defined the expected value, variance, and standard
deviation of a population represented by a discrete probability distribution. Also introduced in this chapter were
bivariate discrete distributions on which an important
application in finance was based. Finally, the two most
important discrete distributions—the binomial and the
Poisson—were presented.
IMPORTANT TERMS
Random variable 218
Discrete random variable 219
Continuous random variable 219
Probability distribution 219
Expected value 223
Bivariate distribution 229
PERT (Project Evaluation and Review Technique) 235
CPM (Critical Path Method) 235
Path 235
Critical path 236
Diversification 237
Binomial experiment 244
Bernoulli process 244
Binomial random variable 244
Binomial probability distribution 245
Cumulative probability 247
Poisson distribution 251
Poisson random variable 251
Poisson experiment 251
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CHAPTER 7
SYMBOLS
Symbol
Pronounced
Represents
ax
Sum of x for all values of x
Summation
n choose x
n!
e
n factorial
Number of combinations
n(n - 1)(n - 2) Á (3)(2)(1)
2.71828 . . .
all x
Cnx
FORMULAS
Expected value (mean)
Laws of expected value and variance for the sum of k
variables, where k Ú 2
E(X) = m = a xP(x)
1. E(X1 + X2 + Á + Xk)
= E(X1) + E(X2) + Á + E(Xk)
2. V(X1 + X2 + Á + Xk)
= V(X1) + V(X2) + Á + V(Xk)
all x
Variance
V(x) = s2 = a (x - m)2P(x)
all x
Standard deviation
if the variables are independent
s = 2s2
Mean and variance of a portfolio of two stocks
Covariance
E(Rp) = w1E(R1) + w2E(R2)
COV(X, Y ) = sxy = a (x - mx)(y - my)P(x, y)
+ 2w1w2COV(R1, R2)
Coefficient of Correlation
= w21s21 + w22s22 + 2w1w2rs1s2
sxy
COV(X, Y )
r =
=
sxsy
sxsy
Mean and variance of a portfolio of k stocks
k
Laws of expected value
E(Rp) = a wiE(Ri)
i=1
1. E(c) = c
k
k
k
V(Rp) = a w2i s2i + 2 a a wiwjCOV(Ri, Rj)
2. E(X + c) = E(X ) + c
i=1
3. E(cX ) = cE(X )
i=1j=i+1
Binomial probability
Laws of variance
P(X = x) =
1. V(c) = 0
2. V(X + c) = V(X )
3. V(cX ) =
V(Rp) = w21V(R1) + w22V(R2)
n!
px(1 - p)n - x
x!(n - x)!
m = np
c2V(X )
s2 = np(1 - p)
Laws of expected value and variance of the sum of two
variables
1. E(X + Y ) = E(X ) + E(Y )
2. V(X + Y ) = V(X ) + V(Y ) + 2COV(X, Y )
s = 2np(1 - p)
Poisson probability
P(X = x) =
e - mmx
x!
COMPUTER INSTRUCTIONS
Probability Distribution
Binomial
Poisson
Excel
Minitab
248
255
249
255
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RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER EXERCISES
7.123 In 2000, Northwest Airlines boasted that 77.4% of
its flights were on time. If we select five Northwest
flights at random, what is the probability that all five
are on time? (Source: Department of Transportation.)
7.124 The final exam in a one-term statistics course is
taken in the December exam period. Students who
are sick or have other legitimate reasons for missing
the exam are allowed to write a deferred exam
scheduled for the first week in January. A statistics
professor has observed that only 2% of all students
legitimately miss the December final exam. Suppose
that the professor has 40 students registered this
term.
a. How many students can the professor expect to
miss the December exam?
b. What is the probability that the professor will
not have to create a deferred exam?
7.125 The number of magazine subscriptions per house-
hold is represented by the following probability distribution.
Magazine subscriptions
per household
0
Probability
.48
1
.35
2
.08
3
.05
4
.04
a. Calculate the mean number of magazine subscriptions per household.
b. Find the standard deviation.
7.126 The number of arrivals at a car wash is Poisson dis-
tributed with a mean of eight per hour.
a. What is the probability that 10 cars will arrive in
the next hour?
b. What is the probability that more than 5 cars will
arrive in the next hour?
c. What is the probability that fewer than 12 cars
will arrive in the next hour?
7.127 The percentage of customers who enter a restaurant
and ask to be seated in a smoking section is 15%.
Suppose that 100 people enter the restaurant.
a. What is the expected number of people who
request a smoking table?
b. What is the standard deviation of the number of
requests for a smoking table?
c. What is the probability that 20 or more people
request a smoking table?
7.128 Lotteries are an important income source for vari-
ous governments around the world. However, the
availability of lotteries and other forms of gambling
have created a social problem: gambling addicts.
A critic of government-controlled gambling contends that 30% of people who regularly buy lottery
tickets are gambling addicts. If we randomly select
10 people among those who report that they regularly buy lottery tickets, what is the probability that
more than 5 of them are addicts?
7.129 The distribution of the number of home runs in
soft-ball games is shown here.
Number of home runs
Probability
0
1
.05 .16
2
3
4
5
.41 .27 .07 .04
a. Calculate the mean number of home runs.
b. Find the standard deviation.
7.130 An auditor is preparing for a physical count of inven-
tory as a means of verifying its value. Items counted
are reconciled with a list prepared by the storeroom
supervisor. In one particular firm, 20% of the items
counted cannot be reconciled without reviewing
invoices. The auditor selects 10 items. Find the probability that 6 or more items cannot be reconciled.
7.131 Shutouts in the National Hockey League occur ran-
domly and independently at a rate of 1 every 20
games. Calculate the probability of the following
events.
a. 2 shutouts in the next 10 games
b. 25 shutouts in 400 games
c. a shutout in to.night’s game
7.132 Most Miami Beach restaurants offer “early-bird”
specials. These are lower-priced meals that are available only from 4 to 6 P.M. However, not all customers who arrive between 4 and 6 P.M. order the
special. In fact, only 70% do.
a. Find the probability that of 80 customers
between 4 and 6 P.M., more than 65 order the
special.
b. What is the expected number of customers who
order the special?
c. What is the standard deviation?
7.133 According to climatologists, the long-term average
for Atlantic storms is 9.6 per season (June 1 to
November 30), with 6 becoming hurricanes and 2.3
becoming intense hurricanes. Find the probability of
the following events.
a. Ten or more Atlantic storms
b. Five or fewer hurricanes
c. Three or more intense hurricanes
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CHAPTER 7
7.134 Researchers at the University of Pennsylvania
School of Medicine theorized that children under 2
years old who sleep in rooms with the light on have
a 40% probability of becoming myopic by age 16.
Suppose that researchers found 25 children who
slept with the light on before they were 2.
a. What is the probability that 10 of them will
become myopic before age 16?
b. What is the probability that fewer than 5 of them
will become myopic before age 16?
c. What is the probability that more than 15 of
them will become myopic before age 16?
7.135 A pharmaceutical researcher working on a cure for
baldness noticed that middle-aged men who are
balding at the crown of their head have a 45% probability of suffering a heart attack over the next
decade. In a sample of 100 middle-age balding men,
what are the following probabilities?
a. More than 50 will suffer a heart attack in the next
decade.
b. Fewer than 44 will suffer a heart attack in the
next decade.
c. Exactly 45 will suffer a heart attack in the next
decade.
7.136 Advertising researchers have developed a theory that
states that commercials that appear in violent television shows are less likely to be remembered and will
thus be less effective. After examining samples of
viewers who watch violent and nonviolent programs
and asking them a series of five questions about the
commercials, the researchers produced the following probability distributions of the number of correct answers.
Viewers of violent shows
x
P (x)
0
1
2
3
4
5
.36
.22
.20
.09
.08
.05
Viewers of nonviolent shows
x
P (x)
0
1
2
3
4
5
.15
.18
.23
.26
.10
.08
a. Calculate the mean and standard deviation of the
number of correct answers among viewers of violent television programs.
b. Calculate the mean and standard deviation of the
number of correct answers among viewers of
nonviolent television programs.
7.137 According to the U.S. census, one-third of all busi-
nesses are owned by women. If we select 25 businesses
at random, what is the probability that 10 or more of
them are owned by women?
7.138 It is recommended that women age 40 and older
have a mammogram annually. A recent report indicated that if a woman has annual mammograms over
a 10-year period, there is a 60% probability that
there will be at least one false-positive result.
(A false-positive mammogram test result is one that
indicates the presence of cancer when, in fact, there
is no cancer.) If the annual test results are independent, what is the probability that in any one year a
mammogram will produce a false-positive result?
(Hint: Find the value of p such that the probability
that a binomial random variable with n ⫽ 10 is
greater than or equal to 1 is .60.)
7.139 In a recent election, the mayor received 60% of the
vote. Last week, a survey was undertaken that asked
100 people whether they would vote for the mayor.
Assuming that her popularity has not changed,
what is the probability that more than 50 people in
the sample would vote for the mayor?
7.140 When Earth traveled through the storm of meteorites
trailing the comet Tempel-Tuttle on November 17,
1998, the storm was 1,000 times as intense as the average meteor storm. Before the comet arrived, telecommunication companies worried about the potential
damage that might be inflicted on the approximately
650 satellites in orbit. It was estimated that each satellite had a 1% chance of being hit, causing damage to
the satellite’s electronic system. One company had five
satellites in orbit at the time. Determine the probability distribution of the number of the company’s satellites that would be damaged.
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CASE 7.1
I
To Bunt or Not to Bunt, That Is
the Question—Part 2
n Case 6.2, we presented the proba-
of runs in situations that are defined
bilities of scoring at least one run
by the number of outs and the bases
and asked you to determine
occupied.
whether the manager should signal for
the batter to sacrifice bunt. The decision was made on the basis of comparing the probability of scoring at least
one run when the manager signaled for
the bunt and when he signaled the
batter to swing away. Another factor
that should be incorporated into the
decision is the number of runs the
manager expects his team to score. In
the same article referred to in Case 6.2,
the author also computed the expected
number of runs scored for each situation. Table 1 lists the expected number
TABLE
1
Expected Number of Runs
Scored
Bases
Occupied
Bases empty
First base
Second base
Third base
First base and
second base
First base and
third base
Second base and
third base
Bases loaded
0 Out 1 Out 2 Outs
.49
.85
1.06
1.21
.27
.52
.69
.82
.10
.23
.34
.38
1.46
1.00
.48
1.65
1.10
.51
1.94
2.31
1.50
1.62
.62
.82
261
© AP Photo/Gene J. Puskar
RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
Assume that the manager wishes to
score as many runs as possible. Using
the same probabilities of the four
outcomes of a bunt listed in Case 6.2,
determine whether the manager should
signal the batter to sacrifice bunt.
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8
© Steve Cole/The Image Bank/Getty Images
8.1
Probability Density Functions
8.2
Normal Distribution
8.3
(Optional) Exponential Distribution
8.4
Other Continuous Distributions
Minimum GMAT Score to Enter Executive
MBA Program
A university has just approved a new Executive MBA Program. The new director believes that to
maintain the prestigious image of the business school, the new program must be seen as having
high standards. Accordingly, the Faculty Council decides that one of the entrance requirements
will be that applicants must score in the top 1% of Graduate Management Admission Test (GMAT)
scores. The director knows that GMAT scores are normally distributed with a mean of 490 and a
© Erik Dreyer/Getty Images
CONTINUOUS PROBABILITY
DISTRIBUTIONS
See page 281.
standard deviation of 61. The only thing she doesn’t know is what the minimum GMAT score for
admission should be.
After introducing the normal distribution, we will return to this question and answer it.
263
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I N T RO D U C T I O N
T
his chapter completes our presentation of probability by introducing continuous
random variables and their distributions. In Chapter 7, we introduced discrete
probability distributions that are employed to calculate the probability associated
with discrete random variables. In Section 7.4, we introduced the binomial distribution,
which allows us to determine the probability that the random variable equals a particular
value (the number of successes). In this way we connected the population represented by
the probability distribution with a sample of nominal data. In this chapter, we introduce
continuous probability distributions, which are used to calculate the probability associated with an interval variable. By doing so, we develop the link between a population and
a sample of interval data.
Section 8.1 introduces probability density functions and uses the uniform density
function to demonstrate how probability is calculated. In Section 8.2, we focus on the
normal distribution, one of the most important distributions because of its role in the
development of statistical inference. Section 8.3 introduces the exponential distribution, a distribution that has proven to be useful in various management-science applications. Finally, in Section 8.4 we introduce three additional continuous distributions.
They will be used in statistical inference throughout the book.
8 . 1 P RO B A B I L I T Y D E N S I T Y F U N C T I O N S
A continuous random variable is one that can assume an uncountable number of values.
Because this type of random variable is so different from a discrete variable, we need to
treat it completely differently. First, we cannot list the possible values because there is
an infinite number of them. Second, because there is an infinite number of values, the
probability of each individual value is virtually 0. Consequently, we can determine the
probability of only a range of values. To illustrate how this is done, consider the histogram we created for the long-distance telephone bills (Example 3.1), which is
depicted in Figure 8.1.
FIGURE
8.1
Histogram for Example 3.1
80
70
Frequency
60
50
40
30
20
10
0
15
30
45
60
75
90
105
120
Long-distance telephone bills
We found, for example, that the relative frequency of the interval 15 to 30 was
37/200. Using the relative frequency approach, we estimate that the probability that a
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CONTINUOUS PROBABILITY DISTRIBUTIONS
265
randomly selected long-distance bill will fall between $15 and $30 is 37/200 ⫽ .185.
We can similarly estimate the probabilities of the other intervals in the histogram.
Interval
Relative Frequency
0 … X … 15
15 6 X … 30
30 6 X … 45
45 6 X … 60
60 6 X … 75
75 6 X … 90
90 6 X … 105
105 6 X … 120
71/200
37/200
13/200
9/200
10/200
18/200
28/200
14/200
Notice that the sum of the probabilities equals 1. To proceed, we set the values along
the vertical axis so that the area in all the rectangles together adds to 1. We accomplish
this by dividing each relative frequency by the width of the interval, which is 15. The
result is a rectangle over each interval whose area equals the probability that the random
variable will fall into that interval.
To determine probabilities of ranges other than the ones created when we drew the
histogram, we apply the same approach. For example, the probability that a longdistance bill will fall between $50 and $80 is equal to the area between 50 and 80 as
shown in Figure 8.2.
FIGURE
8.2
Histogram for Example 3.1: Relative Frequencies Divided by
Interval Width
80/(200 × 15)
70/(200 × 15)
60/(200 × 15)
50/(200 × 15)
40/(200 × 15)
30/(200 × 15)
20/(200 × 15)
10/(200 × 15)
0
15
30
45 50 60
75 80 90
105
120
Long-distance telephone bills
The areas in each shaded rectangle are calculated and added together as follows:
Interval
Height of Rectangle
Base Multiplied by Height
50 6 X … 60
60 6 X … 75
75 6 X … 80
9>1200 * 152 = .00300
10>1200 * 152 = .00333
18>1200 * 152 = .00600
160 - 502 * .00300 = .030
175 - 602 * .00333 = .050
180 - 752 * .00600 = .030
Total ⫽ .110
We estimate that the probability that a randomly selected long-distance bill falls
between $50 and $80 is .11.
If the histogram is drawn with a large number of small intervals, we can smooth the
edges of the rectangles to produce a smooth curve as shown in Figure 8.3. In many cases,
it is possible to determine a function f1x2 that approximates the curve. The function is
called a probability density function. Its requirements are stated in the box.
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FIGURE
8.3
Density Function for Example 3.1
f(x)
x
0
15
30
45
60
75
90
105
120
Requirements for a Probability Density Function
The following requirements apply to a probability density function f1x2
whose range is a … x … b.
1. f1x2 Ú 0 for all x between a and b
2. The total area under the curve between a and b is 1.0
Integral calculus* can often be used to calculate the area under a curve. Fortunately,
the probabilities corresponding to continuous probability distributions that we deal
with do not require this mathematical tool. The distributions will be either simple or
too complex for calculus. Let’s start with the simplest continuous distribution.
Uniform Distribution
To illustrate how we find the area under the curve that describes a probability density
function, consider the uniform probability distribution also called the rectangular
probability distribution.
Uniform Probability Density Function
The uniform distribution is described by the function
f1x2 =
1
b - a
where a … x … b
The function is graphed in Figure 8.4. You can see why the distribution is called rectangular.
*Keller’s website Appendix Continuous Probability Distributions: Calculus Approach demonstrates how
to use integral calculus to determine probabilities and parameters for continuous random variables.
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CONTINUOUS PROBABILITY DISTRIBUTIONS
FIGURE
8.4
267
Uniform Distribution
f(x)
1
––––
b–a
a
b
x
To calculate the probability of any interval, simply find the area under the curve.
For example, to find the probability that X falls between x1 and x2 determine the area in
the rectangle whose base is x2 ⫺ x1 and whose height is 1>1b - a2. Figure 8.5 depicts
the area we wish to find. As you can see, it is a rectangle and the area of a rectangle is
found by multiplying the base times the height.
FIGURE
8.5
P ( x 1<X <x 2)
f(x)
1
––––
b–a
a
x1
x2
b
x
Thus,
P1x1 6 X 6 x22 = Base * Height = 1x2 - x12 *
EXAMPLE
8.1
1
b - a
Uniformly Distributed Gasoline Sales
The amount of gasoline sold daily at a service station is uniformly distributed with a
minimum of 2,000 gallons and a maximum of 5,000 gallons.
a. Find the probability that daily sales will fall between 2,500 and 3,000 gallons.
b. What is the probability that the service station will sell at least 4,000 gallons?
c. What is the probability that the station will sell exactly 2,500 gallons?
SOLUTION
The probability density function is
f1x2 =
1
1
=
5000 - 2000
3000
2000 … x … 5000
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CHAPTER 8
a. The probability that X falls between 2,500 and 3,000 is the area under the curve
between 2,500 and 3,000 as depicted in Figure 8.6a. The area of a rectangle is the
base times the height. Thus,
P12,500 … X … 3,0002 = 13,000 - 2,5002 * a
b. P1X Ú 4,0002 = 15,000 - 4,0002 * a
1
b = .1667
3,000
1
b = .3333
3,000
[See Figure 8.6b.]
c. P1X = 2,5002 = 0
Because there is an uncountable infinite number of values of X, the probability of
each individual value is zero. Moreover, as you can see from Figure 8.6c, the area
of a line is 0.
Because the probability that a continuous random variable equals any individual value is 0, there is no difference between P12,500 … X … 3,0002 and
P(2,500 ⬍ X ⬍ 3,000). Of course, we cannot say the same
thing about discrete random variables.
FIGURE
8.6
Density Functions for Example 8.1
f(x)
1
–––––
3,000
x
2,000 2,500 3,000
5,000
(a) P(2,500 < X < 3,000)
f(x)
1
–––––
3,000
x
2,000
4,000
5,000
(b) P(4,000 < X < 5,000)
f(x)
1
–––––
3,000
x
2,000 2,500
5,000
(c) P(X = 2,500)
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CONTINUOUS PROBABILITY DISTRIBUTIONS
269
Using a Continuous Distribution to Approximate a Discrete
Distribution
In our definition of discrete and continuous random variables, we distinguish between them
by noting whether the number of possible values is countable or uncountable. However, in
practice, we frequently use a continuous distribution to approximate a discrete one when
the number of values the variable can assume is countable but large. For example, the number of possible values of weekly income is countable. The values of weekly income
expressed in dollars are 0, .01, .02, . . . . Although there is no set upper limit, we can easily
identify (and thus count) all the possible values. Consequently, weekly income is a discrete
random variable. However, because it can assume such a large number of values, we prefer
to employ a continuous probability distribution to determine the probability associated
with such variables. In the next section, we introduce the normal distribution, which is
often used to describe discrete random variables that can assume a large number of values.
EXERCISES
8.1 Refer to Example 3.2. From the histogram for
investment A, estimate the following probabilities.
a. P1X 7 452
b. P110 6 X 6 402
c. P1X 6 252
d. P135 6 X 6 652
8.2 Refer to Example 3.2. Estimate the following from
the histogram of the returns on investment B.
a. P1X 7 452
b. P110 6 X 6 402
c. P1X 6 252
d. P135 6 X 6 652
8.3 Refer to Example 3.3. From the histogram of the
marks, estimate the following probabilities.
a. P155 6 X 6 802
b. P1X 7 652
c. P1X 6 852
d. P175 6 X 6 852
8.4 A random variable is uniformly distributed between
5 and 25.
a. Draw the density function.
b. Find P1X 7 252.
c. Find P110 6 X 6 152.
d. Find P15.0 6 X 6 5.12.
8.5 A uniformly distributed random variable has mini-
mum and maximum values of 20 and 60, respectively.
a. Draw the density function.
b. Determine P135 6 X 6 452.
c. Draw the density function including the calculation of the probability in part (b).
8.6 The amount of time it takes for a student to complete
a statistics quiz is uniformly distributed between
30 and 60 minutes. One student is selected at random.
Find the probability of the following events.
a. The student requires more than 55 minutes to
complete the quiz.
b. The student completes the quiz in a time
between 30 and 40 minutes.
c. The student completes the quiz in exactly 37.23
minutes.
8.7 Refer to Exercise 8.6. The professor wants to reward
(with bonus marks) students who are in the lowest
quarter of completion times. What completion time
should he use for the cutoff for awarding bonus marks?
8.8 Refer to Exercise 8.6. The professor would like to
track (and possibly help) students who are in the top
10% of completion times. What completion time
should he use?
8.9 The weekly output of a steel mill is a uniformly dis-
tributed random variable that lies between 110 and
175 metric tons.
a. Compute the probability that the steel mill will
produce more than 150 metric tons next week.
b. Determine the probability that the steel mill will
produce between 120 and 160 metric tons next
week.
8.10 Refer to Exercise 8.9. The operations manager
labels any week that is in the bottom 20% of production a “bad week.” How many metric tons should be
used to define a bad week?
8.11 A random variable has the following density function.
f1x2 = 1 - .5x
0 6 x 6 2
a. Graph the density function.
b. Verify that f1x2 is a density function.
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a.
b.
c.
d.
e.
c. Find P1X 7 12.
d. Find P1X 6 .52.
e. Find P1X = 1.52.
8.12 The following function is the density function for
the random variable X:
Graph the density function.
Find the probability that X lies between 1 and 3.
What is the probability that X lies between 4 and 8?
Compute the probability that X is less than 7.
Find the probability that X is greater than 3.
8.14 The following is a graph of a density function.
x - 1
1 6 x 6 5
8
a. Graph the density function.
b. Find the probability that X lies between 2 and 4.
c. What is the probability that X is less than 3?
f1x2 =
f(x)
.10
8.13 The following density function describes the ran-
dom variable X.
x
25
f1x2 = d
10 - x
25
x
0
0 6 x 6 5
20
a. Determine the density function.
b. Find the probability that X is greater than 10.
c. Find the probability that X lies between 6 and 12.
5 6 x 6 10
8 . 2 N OR M A L D I S T R I B U T I O N
The normal distribution is the most important of all probability distributions because
of its crucial role in statistical inference.
Normal Density Function
The probability density function of a normal random variable is
1
f1x2 =
s22p
1 x-m 2
b
- a
e 2 s
-q 6 x 6 q
where e ⫽ 2.71828 . . . and ⫽ 3.14159 . . .
Figure 8.7 depicts a normal distribution. Notice that the curve is symmetric about
its mean and the random variable ranges between ⫺⬁ and ⫹⬁.
FIGURE
8.7
Normal Distribution
f(x)
m
x
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CONTINUOUS PROBABILITY DISTRIBUTIONS
271
The normal distribution is described by two parameters, the mean and the standard deviation . In Figure 8.8, we demonstrate the effect of changing the value of .
Obviously, increasing shifts the curve to the right and decreasing shifts it to the left.
FIGURE
8.8
Normal Distributions with the Same Variance but Different Means
f(x)
x
2
4
6
Figure 8.9 describes the effect of . Larger values of widen the curve and smaller
ones narrow it.
FIGURE
8.9
Normal Distributions with the Same Means but Different Variances
f(x)
s = 10
s = 12
s = 15
x
S E E I N G S TAT I S T I C S
applet 4 Normal Distribution Parameters
This applet can be used to see the effect
you increase or decrease the standard
of changing the values of the mean and
deviation.
standard deviation of a normal
distribution.
Move the top slider left or right to
decrease or increase the mean of the
distribution. Notice that when you
change the value of the mean, the
shape stays the same; only the location
changes. Move the second slider to
change the standard deviation. The
shape of the bell curve is changed when
Applet Exercises
4.1 Move the slider bar for the standard
deviation so that the standard
deviation of the red distribution is
greater than 1. What does this do
to the spread of the normal
distribution? Does it squeeze it or
deviation of the red distribution is
stretch it?
less than 1. What does this do to the
4.2 Move the slider bar for the standard
deviation so that the standard
spread of the normal distribution?
Does it squeeze it or stretch it?
(Continued)
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4.3 Move both the mean and standard
values to slide the red distribution
squeeze or stretch the red
deviation sliders so that the red
back (forward) so that the centers
distribution so that it would have
distribution is different from the
of the red and blue distributions
the same spread as the blue
blue distribution. What would you
would overlap? By what would you
distribution?
need to subtract from the red
need to divide the red values to
Calculating Normal Probabilities
To calculate the probability that a normal random variable falls into any interval, we must
compute the area in the interval under the curve. Unfortunately, the function is not as
simple as the uniform precluding the use of simple mathematics or even integral calculus.
Instead we will resort to using a probability table similar to Tables 1 and 2 in Appendix B,
which are used to calculate binomial and Poisson probabilities, respectively. Recall that to
determine binomial probabilities from Table 1 we needed probabilities for selected values
of n and p. Similarly, to find Poisson probabilities we needed probabilities for each value
of that we chose to include in Table 2. It would appear then that we will need a separate
table for normal probabilities for a selected set of values of and . Fortunately, this
won’t be necessary. Instead, we reduce the number of tables needed to one by standardizing the random variable. We standardize a random variable by subtracting its mean and
dividing by its standard deviation. When the variable is normal, the transformed variable
is called a standard normal random variable and denoted by Z; that is,
Z =
X - m
s
The probability statement about X is transformed by this formula into a statement
about Z. To illustrate how we proceed, consider the following example.
EXAMPLE
8. 2
Normally Distributed Gasoline Sales
Suppose that the daily demand for regular gasoline at another gas station is normally
distributed with a mean of 1,000 gallons and a standard deviation of 100 gallons. The
station manager has just opened the station for business and notes that there is exactly
1,100 gallons of regular gasoline in storage. The next delivery is scheduled later today
at the close of business. The manager would like to know the probability that he will
have enough regular gasoline to satisfy today’s demands.
SOLUTION
The amount of gasoline on hand will be sufficient to satisfy demand if the demand is
less than the supply. We label the demand for regular gasoline as X, and we want to find
the probability
P1X … 1,1002
Note that because X is a continuous random variable, we can also express the probability as
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CONTINUOUS PROBABILITY DISTRIBUTIONS
273
P1X 6 1,1002
because the area for X ⫽ 1,100 is 0.
Figure 8.10 describes a normal curve with mean of 1,000 and standard deviation of
100, and the area we want to find.
FIGURE
8.10
P ( X <1,100)
x
1,000
1,100
The first step is to standardize X. However, if we perform any operations on X, we
must perform the same operations on 1,100. Thus,
P1X 6 1,1002 = P ¢
X - m
1,100 - 1,000
6
≤ = P1Z 6 1.002
s
100
Figure 8.11 describes the transformation that has taken place. Notice that the variable X was transformed into Z, and 1,100 was transformed into 1.00. However, the area
has not changed. In other words, the probability that we wish to compute P(X ⬍ 1,100)
is identical to P(Z ⬍ 1.00).
FIGURE
8.11
P ( Z <1.00)
z
0
1.00
The values of Z specify the location of the corresponding value of X. A value of
Z ⫽ 1 corresponds to a value of X that is 1 standard deviation above the mean. Notice
as well that the mean of Z, which is 0, corresponds to the mean of X.
If we know the mean and standard deviation of a normally distributed random variable, we can always transform the probability statement about X into a probability
statement about Z. Consequently, we need only one table, Table 3 in Appendix B, the
standard normal probability table, which is reproduced here as Table 8.1.*
*In previous editions we have used another table, which lists P(0 ⬍ Z ⬍ z). Supplementary Appendix
Determining Normal Probabilities using P(0 ⬍ Z ⬍ z) provides instructions and examples using this
table.
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Normal Probabilities (Table 3 in Appendix B)
Z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
⫺3.0
⫺2.9
⫺2.8
⫺2.7
⫺2.6
⫺2.5
⫺2.4
⫺2.3
⫺2.2
⫺2.1
⫺2.0
⫺1.9
⫺1.8
⫺1.7
⫺1.6
⫺1.5
⫺1.4
⫺1.3
⫺1.2
⫺1.1
⫺1.0
⫺0.9
⫺0.8
⫺0.7
⫺0.6
⫺0.5
⫺0.4
⫺0.3
⫺0.2
⫺0.1
⫺0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
0.0013
0.0019
0.0026
0.0035
0.0047
0.0062
0.0082
0.0107
0.0139
0.0179
0.0228
0.0287
0.0359
0.0446
0.0548
0.0668
0.0808
0.0968
0.1151
0.1357
0.1587
0.1841
0.2119
0.2420
0.2743
0.3085
0.3446
0.3821
0.4207
0.4602
0.5000
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.0013
0.0018
0.0025
0.0034
0.0045
0.0060
0.0080
0.0104
0.0136
0.0174
0.0222
0.0281
0.0351
0.0436
0.0537
0.0655
0.0793
0.0951
0.1131
0.1335
0.1562
0.1814
0.2090
0.2389
0.2709
0.3050
0.3409
0.3783
0.4168
0.4562
0.4960
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.0013
0.0018
0.0024
0.0033
0.0044
0.0059
0.0078
0.0102
0.0132
0.0170
0.0217
0.0274
0.0344
0.0427
0.0526
0.0643
0.0778
0.0934
0.1112
0.1314
0.1539
0.1788
0.2061
0.2358
0.2676
0.3015
0.3372
0.3745
0.4129
0.4522
0.4920
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.0012
0.0017
0.0023
0.0032
0.0043
0.0057
0.0075
0.0099
0.0129
0.0166
0.0212
0.0268
0.0336
0.0418
0.0516
0.0630
0.0764
0.0918
0.1093
0.1292
0.1515
0.1762
0.2033
0.2327
0.2643
0.2981
0.3336
0.3707
0.4090
0.4483
0.4880
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.0012
0.0016
0.0023
0.0031
0.0041
0.0055
0.0073
0.0096
0.0125
0.0162
0.0207
0.0262
0.0329
0.0409
0.0505
0.0618
0.0749
0.0901
0.1075
0.1271
0.1492
0.1736
0.2005
0.2296
0.2611
0.2946
0.3300
0.3669
0.4052
0.4443
0.4840
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.0011
0.0016
0.0022
0.0030
0.0040
0.0054
0.0071
0.0094
0.0122
0.0158
0.0202
0.0256
0.0322
0.0401
0.0495
0.0606
0.0735
0.0885
0.1056
0.1251
0.1469
0.1711
0.1977
0.2266
0.2578
0.2912
0.3264
0.3632
0.4013
0.4404
0.4801
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.0011
0.0015
0.0021
0.0029
0.0039
0.0052
0.0069
0.0091
0.0119
0.0154
0.0197
0.0250
0.0314
0.0392
0.0485
0.0594
0.0721
0.0869
0.1038
0.1230
0.1446
0.1685
0.1949
0.2236
0.2546
0.2877
0.3228
0.3594
0.3974
0.4364
0.4761
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.0011
0.0015
0.0021
0.0028
0.0038
0.0051
0.0068
0.0089
0.0116
0.0150
0.0192
0.0244
0.0307
0.0384
0.0475
0.0582
0.0708
0.0853
0.1020
0.1210
0.1423
0.1660
0.1922
0.2206
0.2514
0.2843
0.3192
0.3557
0.3936
0.4325
0.4721
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.0010
0.0014
0.0020
0.0027
0.0037
0.0049
0.0066
0.0087
0.0113
0.0146
0.0188
0.0239
0.0301
0.0375
0.0465
0.0571
0.0694
0.0838
0.1003
0.1190
0.1401
0.1635
0.1894
0.2177
0.2483
0.2810
0.3156
0.3520
0.3897
0.4286
0.4681
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.0010
0.0014
0.0019
0.0026
0.0036
0.0048
0.0064
0.0084
0.0110
0.0143
0.0183
0.0233
0.0294
0.0367
0.0455
0.0559
0.0681
0.0823
0.0985
0.1170
0.1379
0.1611
0.1867
0.2148
0.2451
0.2776
0.3121
0.3483
0.3859
0.4247
0.4641
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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CONTINUOUS PROBABILITY DISTRIBUTIONS
275
This table is similar to the ones we used for the binomial and Poisson distributions;
that is, this table lists cumulative probabilities
P1Z 6 z2
for values of z ranging from ⫺3.09 to ⫹ 3.09
To use the table, we simply find the value of z and read the probability. For example, the probability P1Z 6 2.002 is found by finding 2.0 in the left margin and under
the heading .00 finding .9772. The probability P1Z 6 2.012 is found in the same row
but under the heading .01. It is .9778.
Returning to Example 8.2, the probability we seek is found in Table 8.1 by finding
1.0 in the left margin. The number to its right under the heading .00 is .8413.
See Figure 8.12.
FIGURE
8.12
P ( Z <1.00)
z
.00
.01
.02
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212
1.0
.8413
.8438
.8461
1.1
.8643
.8665
.8686
1.2
.8849
.8869
.8888
z
0
1.0
As was the case with Tables 1 and 2, we can also determine the probability that the
standard normal random variable is greater than some value of z. For example, we find
the probability that Z is greater than 1.80 by determining the probability that Z is less
than 1.80 and subtracting that value from 1. Applying the complement rule, we get
P1Z 7 1.802 = 1 - P1Z 6 1.802 = 1 - .9641 = .0359
See Figure 8.13.
FIGURE
8.13
P ( Z >1.80)
z
.00
.01
.02
1.6
.9452
.9463
.9474
1.7
.9554
.9564
.9573
1.8
.9641
.9649
.9656
1.9
.9713
.9719
.9726
2.0
.9772
.9778
.9783
z
0
1.80
We can also easily determine the probability that a standard normal random
variable lies between two values of z. For example, we find the probability
P1 -0.71 6 Z 6 0.922
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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CHAPTER 8
by finding the two cumulative probabilities and calculating their difference; that is,
P1Z 6 - 0.712 = .2389
and
P1Z 6 0.922 = .8212
Hence,
P1- 0.71 6 Z 6 0.922 = P1Z 6 922 - P1Z 6 - 0.712 = .8212 - .2389 = .5823
Figure 8.14 depicts this calculation.
FIGURE
8.14
P( 0.71< Z <0.92)
z
.00
.01
.02
−0.8
.2119
.2090
.2061
−0.7
.2420
.2389
.2358
−0.6
.2743
.2709
.2676
−0.5
.3085
.3050
.3015
−0.4
.3446
.3409
.3372
−0.3
.3821
.3783
.3745
−0.2
.4207
.4168
.4129
−0.1
.4602
.4562
.4522
−0.0
.5000
.4960
.4920
0.0
.5000
.5040
.5080
0.1
.5398
.5438
.5478
0.2
.5793
.5832
.5871
0.3
.6179
.6217
.6255
0.4
.6554
.6591
.6628
0.5
.6915
.6950
.6985
0.6
.7257
.7291
.7324
0.7
.7580
.7611
.7642
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212
1.0
.8413
.8438
.8461
z
–0.71
0
0.92
Notice that the largest value of z in the table is 3.09, and that
P1Z 6 3.092 = .9990. This means that
P1Z 7 3.092 = 1 - .9990 = .0010
However, because the table lists no values beyond 3.09, we approximate any area
beyond 3.10 as 0. In other words,
P1Z 7 3.102 = P1Z 6 - 3.102 L 0
Recall that in Tables 1 and 2 we were able to use the table to find the probability
that X is equal to some value of x, but we won’t do the same with the normal table.
Remember that the normal random variable is continuous and the probability that a
continuous random variable is equal to any single value is 0.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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CONTINUOUS PROBABILITY DISTRIBUTIONS
277
S E E I N G S TAT I S T I C S
applet 5 Normal Distribution Areas
This applet can be used to show the
the mean and standard deviation
calculation of the probability of any
correspond to a mean of 50 and a
interval for any values of and . Click
standard deviation of 10.
or drag anywhere in the graph to move
the nearest end to that point. Adjust the
Applet Exercises
ends to correspond to either z-scores or
The graph is initially set with mean 50
actual scores. The area under the normal
and standard deviation 10. Change it
curve between the two endpoints is
so that it represents the distribution of
highlighted in red. The size of this area
IQs, which are normally distributed with
corresponds to the probability of
a mean of 100 and a standard deviation
scores are within one standard
obtaining a score between the two
of 16.
deviation of the mean?
endpoints. You can change the mean
5.1 About what proportion of people have
and standard deviation of the actual
IQ scores equal to or less than 116?
scores by changing the numbers in the
5.2 About what proportion of people
text boxes. After changing a number,
press the Enter or Return key to update
have IQ scores between 100 and 116?
5.3 About what proportion have IQ
the graph. When this page first loads,
scores greater than 120?
5.4 About what proportion of the
5.5 About what proportion of the
scores are within two standard
deviations of the mean?
5.6 About what proportion of the
scores are within three standard
deviations of the mean?
APPLICATIONS in FINANCE
Measuring Risk
In previous chapters, we discussed several probability and statistical applications
© Davis Barber/PhotoEdit
in finance where we wanted to measure and perhaps reduce the risk associated
with investments. In Example 3.2, we drew histograms to gauge the spread of
the histogram of the returns on two investments. We repeated this example in
Chapter 4, where we computed the standard deviation and variance as numerical
measures of risk. In Section 7.3, we developed an important application in finance
in which we emphasized reducing the variance of the returns on a portfolio. However,
we have not demonstrated why risk is measured by the variance and standard deviation.
The following example corrects this deficiency.
EXAMPLE
8.3
Probability of a Negative Return on Investment
Consider an investment whose return is normally distributed with a mean of 10% and a
standard deviation of 5%.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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a. Determine the probability of losing money.
b. Find the probability of losing money when the standard deviation is equal
to 10%.
SOLUTION
a. The investment loses money when the return is negative. Thus, we wish to determine
P1X 6 02
The first step is to standardize both X and 0 in the probability statement:
P1X 6 02 = Pa
X - m
0 - 10
6
b = P1Z 6 - 2.002 = .0228
s
5
Therefore, the probability of losing money is .0228.
b. If we increase the standard deviation to 10%, the probability of suffering a loss
becomes
P1X 6 02 = Pa
X - m
0 - 10
6
b = P1Z 6 - 1.002 = .1587
s
10
As you can see, increasing the standard deviation increases the probability of
losing money. Note that increasing the standard deviation will also increase
the probability that the return will exceed some relatively large amount.
However, because investors tend to be risk averse, we emphasize the increased
probability of negative returns when discussing the effect of increasing the
standard deviation.
Finding Values of Z
There is a family of problems that require us to determine the value of Z given a probability. We use the notation ZA to represent the value of z such that the area to its right
under the standard normal curve is A; that is, ZA is a value of a standard normal random
variable such that
P1Z 7 ZA2 = A
Figure 8.15 depicts this notation.
FIGURE
8.15
P ( Z > Z A )= A
A
0
zA
z
To find ZA for any value of A requires us to use the standard normal table backward. As you saw in Example 8.2, to find a probability about Z, we must find the value
of z in the table and determine the probability associated with it. To use the table backward, we need to specify a probability and then determine the z-value associated with it.
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CONTINUOUS PROBABILITY DISTRIBUTIONS
279
We’ll demonstrate by finding Z.025. Figure 8.16 depicts the standard normal curve and
Z.025. Because of the format of the standard normal table, we begin by determining the
area less than Z.025, which is 1 ⫺ .025 ⫽ .9750. (Notice that we expressed this probability with four decimal places to make it easier for you to see what you need to do.)
We now search through the probability part of the table looking for .9750. When we
locate it, we see that the z-value associated with it is 1.96.
Thus, Z.025 ⫽ 1.96, which means that P1Z 7 1.962 = .025.
FIGURE
8.16
Z .025
1 – .025 = .9750
.025
z.025
0
EXAMPLE
8.4
z
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
1.0
.8413
.8438
.8461
.8485
.8508
.8531
.8554
.8577
.8599
.8621
1.1
.8643
.8665
.8686
.8708
.8729
.8749
.8770
.8790
.8810
.8830
1.2
.8849
.8869
.8888
.8907
.8925
.8944
.8962
.8980
.8997
.9015
1.3
.9032
.9049
.9066
.9082
.9099
.9115
.9131
.9147
.9162
.9177
1.4
.9192
.9207
.9222
.9236
.9251
.9265
.9279
.9292
.9306
.9319
1.5
.9332
.9345
.9357
.9370
.9382
.9394
.9406
.9418
.9429
.9441
1.6
.9452
.9463
.9474
.9484
.9495
.9505
.9515
.9525
.9535
.9545
1.7
.9554
.9564
.9573
.9582
.9591
.9599
.9608
.9616
.9625
.9633
1.8
.9641
.9649
.9656
.9664
.9671
.9678
.9686
.9693
.9699
.9706
1.9
.9713
.9719
.9726
.9732
.9738
.9744
.9750
.9756
.9761
.9767
2.0
.9772
.9778
.9783
.9788
.9793
.9798
.9803
.9808
.9812
.9817
2.1
.9821
.9826
.9830
.9834
.9838
.9842
.9846
.9850
.9854
.9857
2.2
.9861
.9864
.9868
.9871
.9875
.9878
.9881
.9884
.9887
.9890
2.3
.9893
.9896
.9898
.9901
.9904
.9906
.9909
.9911
.9913
.9916
2.4
.9918
.9920
.9922
.9925
.9927
.9929
.9931
.9932
.9934
.9936
2.5
.9938
.9940
.9941
.9943
.9945
.9946
.9948
.9949
.9951
.9952
2.6
.9953
.9955
.9956
.9957
.9959
.9960
.9961
.9962
.9963
.9964
2.7
.9965
.9966
.9967
.9968
.9969
.9970
.9971
.9972
.9973
.9974
2.8
.9974
.9975
.9976
.9977
.9977
.9978
.9979
.9979
.9980
.9981
2.9
.9981
.9982
.9982
.9983
.9984
.9984
.9985
.9985
.9986
.9986
3.0
.9987
.9987
.9987
.9988
.9988
.9989
.9989
.9989
.9990
.9990
Finding Z.05
Find the value of a standard normal random variable such that the probability that the
random variable is greater than it is 5%.
SOLUTION
We wish to determine Z.05. Figure 8.17 depicts the normal curve and Z.05. If .05 is the
area in the tail, then the probability less than Z.05 must be 1 ⫺ .05 ⫽ .9500. To find Z.05
we search the table looking for the probability .9500. We don’t find this probability, but
we find two values that are equally close: .9495 and .9505. The Z-values associated with
these probabilities are 1.64 and 1.65, respectively. The average is taken as Z.05. Thus,
Z.05 ⫽ 1.645.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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FIGURE
8.17
Z .05
1 – .05 = .9500
.05
EXAMPLE
8.5
z
z.05
0
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
1.0
.8413
.8438
.8461
.8485
.8508
.8531
.8554
.8577
.8599
.8621
1.1
.8643
.8665
.8686
.8708
.8729
.8749
.8770
.8790
.8810
.8830
1.2
.8849
.8869
.8888
.8907
.8925
.8944
.8962
.8980
.8997
.9015
1.3
.9032
.9049
.9066
.9082
.9099
.9115
.9131
.9147
.9162
.9177
1.4
.9192
.9207
.9222
.9236
.9251
.9265
.9279
.9292
.9306
.9319
1.5
.9332
.9345
.9357
.9370
.9382
.9394
.9406
.9418
.9429
.9441
1.6
.9452
.9463
.9474
.9484
.9495
.9505
.9515
.9525
.9535
.9545
1.7
.9554
.9564
.9573
.9582
.9591
.9599
.9608
.9616
.9625
.9633
1.8
.9641
.9649
.9656
.9664
.9671
.9678
.9686
.9693
.9699
.9706
1.9
.9713
.9719
.9726
.9732
.9738
.9744
.9750
.9756
.9761
.9767
2.0
.9772
.9778
.9783
.9788
.9793
.9798
.9803
.9808
.9812
.9817
2.1
.9821
.9826
.9830
.9834
.9838
.9842
.9846
.9850
.9854
.9857
2.2
.9861
.9864
.9868
.9871
.9875
.9878
.9881
.9884
.9887
.9890
2.3
.9893
.9896
.9898
.9901
.9904
.9906
.9909
.9911
.9913
.9916
2.4
.9918
.9920
.9922
.9925
.9927
.9929
.9931
.9932
.9934
.9936
2.5
.9938
.9940
.9941
.9943
.9945
.9946
.9948
.9949
.9951
.9952
2.6
.9953
.9955
.9956
.9957
.9959
.9960
.9961
.9962
.9963
.9964
2.7
.9965
.9966
.9967
.9968
.9969
.9970
.9971
.9972
.9973
.9974
2.8
.9974
.9975
.9976
.9977
.9977
.9978
.9979
.9979
.9980
.9981
2.9
.9981
.9982
.9982
.9983
.9984
.9984
.9985
.9985
.9986
.9986
3.0
.9987
.9987
.9987
.9988
.9988
.9989
.9989
.9989
.9990
.9990
Finding ⫺Z.05
Find the value of a standard normal random variable such that the probability that the
random variable is less than it is 5%.
SOLUTION
Because the standard normal curve is symmetric about 0, we wish to find ⫺Z.05. In
Example 8.4 we found Z.05 ⫽ 1.645. Thus, ⫺ Z.05 ⫽ ⫺ 1.645. See Figure 8.18.
FIGURE
8.18
-Z .05
.05
.05
z
–z.05 = –1.645
0
z.05 = 1.645
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Minimum GMAT Score to Enter Executive MBA
Program: Solution
Figure 8.19 depicts the distribution of GMAT scores. We’ve labeled the minimum score needed to
enter the new MBA program X.01 such that
P1X 7 X.012 = .01
FIGURE
8.19
281
© Erik Dreyer/Getty Images
CONTINUOUS PROBABILITY DISTRIBUTIONS
Minimum GMAT Score
z
.00
.01
.02
.03
.04
1.0
.8413
.8438
.8461
.8485
.8508
1.1
.8643
.8665
.8686
.8708
.8729
1.2
.8849
.8869
.8888
.8907
.8925
1.3
.9032
.9049
.9066
.9082
.9099
1.4
.9192
.9207
.9222
.9236
.9251
1.5
.9332
.9345
.9357
.9370
.9382
1.6
.9452
.9463
.9474
.9484
.9495
1.7
.9554
.9564
.9573
.9582
.9591
1.8
.9641
.9649
.9656
.9664
.9671
1.9
.9713
.9719
.9726
.9732
.9738
2.0
.9772
.9778
.9783
.9788
.9793
2.1
.9821
.9826
.9830
.9834
.9838
2.2
.9861
.9864
.9868
.9871
.9875
2.3
.9893
.9896
.9898
.9901
.9904
2.4
.9918
.9920
.9922
.9925
.9927
2.5
.9938
.9940
.9941
.9943
.9945
2.6
.9953
.9955
.9956
.9957
.9959
2.7
.9965
.9966
.9967
.9968
.9969
2.8
.9974
.9975
.9976
.9977
.9977
2.9
.9981
.9982
.9982
.9983
.9984
3.0
.9987
.9987
.9987
.9988
.9988
1 – .01 = .9900
.01
0
z
z.01
.01
490
x.01
GMAT
Above the normal curve, we depict the standard normal curve and Z.01. We can determine the value of Z.01 as we did in Example 8.4.
In the standard normal table, we find 1 ⫺ .01 ⫽ .9900 (its closest value in the table is .9901) and the Z-value 2.33. Thus, the standardized value of X.01 is Z.01 ⫽ 2.33. To find X.01, we must unstandardize Z.01. We do so by solving for X.01 in the equation
Z.01 =
X.01 - m
s
Substituting Z.01 ⫽ 2.33, ⫽ 490, and ⫽ 61, we find
2.33 =
X.01 - 490
61
Solving, we get
X.01 = 2.331612 + 490 = 632.13
Rounding up (GMAT scores are integers), we find that the minimum GMAT score to enter the Executive MBA Program is 633.
ZA and Percentiles
In Chapter 4, we introduced percentiles, which are measures of relative standing. The values of ZA are the 100(1 ⫺ A)th percentiles of a standard normal random variable. For
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CHAPTER 8
example, Z.05 ⫽ 1.645, which means that 1.645 is the 95th percentile: 95% of all values of Z
are below it, and 5% are above it. We interpret other values of ZA similarly.
Using the Computer
EXCEL
INSTRUCTIONS
We can use Excel to compute probabilities as well as values of X and Z. To compute
cumulative normal probabilities P1X 6 x2, type (in any cell)
= NORMDIST13X4, 3m4, 3s4,True2
(Typing “True” yields a cumulative probability. Typing “False” will produce the value of
the normal density function, a number with little meaning.)
If you type 0 for and 1 for , you will obtain standard normal probabilities.
Alternatively, type
NORMSDIST instead of NORMDIST and enter the value of z.
In Example 8.2 we found P1X 6 1,1002 = P1Z 6 1.002 = .8413. To instruct Excel
to calculate this probability, we enter
= NORMDIST11100, 1000, 100, True2
or
= NORMSDIST11.002
To calculate a value for ZA, type
= NORMSINV131 - A42
In Example 8.4, we would type
= NORMSINV1.952
and produce 1.6449. We calculated Z.05 ⫽ 1.645.
To calculate a value of x given the probability P1X 7 x2 = A, enter
= NORMINV11 - A, m, s2
The chapter-opening example would be solved by typing
= NORMINV1.99, 490, 612
which yields 632.
MINITAB
INSTRUCTIONS
We can use Minitab to compute probabilities as well as values of X and Z.
Check Calc, Probability Distributions, and Normal . . . and either Cumulative probability [to determine P1X 6 x2] or Inverse cumulative probability to find the value of x.
Specify the Mean and Standard deviation.
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283
APPLICATIONS in OPERATIONS MANAGEMENT
Inventory Management
Every organization maintains some inventory, which is defined as a stock of items.
© John Zoiner/Workbook
Stock/Jupiterimages
For example, grocery stores hold inventories of almost all the products they sell.
When the total number of products drops to a specified level, the manager
arranges for the delivery of more products. An automobile repair shop keeps an
inventory of a large number of replacement parts. A school keeps stock of items
that it uses regularly, including chalk, pens, envelopes, file folders, and paper clips.
There are costs associated with inventories.
These include the cost of capital, losses (theft and obsolescence), and warehouse space, as
well as maintenance and record keeping. Management scientists have developed many models to
help determine the optimum inventory level that balances the cost of inventory with the cost of
shortages and the cost of making many small orders. Several of these models are deterministic—
that is, they assume that the demand for the product is constant. However, in most realistic situations, the demand is a random variable. One commonly applied probabilistic model assumes that
the demand during lead time is a normally distributed random variable. Lead time is defined as
the amount of time between when the order is placed and when it is delivered.
The quantity ordered is usually calculated by attempting to minimize the total costs, including the cost of ordering and the cost of maintaining inventory. (This topic is discussed in most
management-science courses.) Another critical decision involves the reorder point, which is the
level of inventory at which an order is issued to its supplier. If the reorder point is too low, the
company will run out of product, suffering the loss of sales and potentially customers who will
go to a competitor. If the reorder point is too high, the company will be carrying too much inventory, which costs money to buy and store. In some companies, inventory has a tendency to walk
out the back door or become obsolete. As a result, managers create a safety stock, which is the
extra amount of inventory to reduce the times when the company has a shortage. They do so by
setting a service level, which is the probability that the company will not experience a shortage.
The method used to determine the reorder point is be demonstrated with Example 8.6.
EXAMPLE
8.6
Determining the Reorder Point
During the spring, the demand for electric fans at a large home-improvement store is
quite strong. The company tracks inventory using a computer system so that it knows
how many fans are in the inventory at any time. The policy is to order a new shipment
of 250 fans when the inventory level falls to the reorder point, which is 150. However,
this policy has resulted in frequent shortages and thus lost sales because both lead time
and demand are highly variable. The manager would like to reduce the incidence of
shortages so that only 5% of orders will arrive after inventory drops to 0 (resulting in a
shortage). This policy is expressed as a 95% service level. From previous periods, the
company has determined that demand during lead time is normally distributed with a
mean of 200 and a standard deviation of 50. Find the reorder point.
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SOLUTION
The reorder point is set so that the probability that demand during lead time exceeds
this quantity is 5%. Figure 8.20 depicts demand during lead time and the reorder point.
As we did in the solution to the chapter-opening example, we find the standard normal
value such that the area to its right is .05. The standardized value of the reorder point is
Z.05 ⫽ 1.645. To find the reorder point (ROP), we must unstandardize Z.05.
ROP - m
s
ROP - 200
1.645 =
50
ROP = 5011.6452 + 200 = 282.25
Z.05 =
which we round up to 283. The policy is to order a new batch of fans when there are
283 fans left in inventory.
FIGURE
8.20
Distribution of Demand During Lead Time
.05
200
ROP
Demand during
lead time
EXERCISES
In Exercises 8.15 to 8.30, find the probabilities.
8.15 P1Z 6 1.502
8.16 P1Z 6 1.512
8.17 P1Z 6 1.552
8.18 P1Z 6 - 1.592
8.19 P1Z 6 - 1.602
8.20 P1Z 6 - 2.302
8.21 P1 - 1.40 6 Z 6 .602
8.22 P1Z 7 - 1.442
8.23 P1Z 6 2.032
8.24 P1Z 7 1.672
8.25 P1Z 6 2.842
8.26 P11.14 6 Z 6 2.432
8.27 P1 - 0.91 6 Z 6 - 0.332
8.28 P1Z 7 3.092
8.29 P1Z 7 02
8.30 P1Z 7 4.02
8.31 Find z.02.
8.32 Find z.045.
8.33 Find z.20.
8.34 X is normally distributed with mean 100 and stan-
dard deviation 20. What is the probability that X is
greater than 145?
8.35 X is normally distributed with mean 250 and stan-
dard deviation 40. What value of X does only the top
15% exceed?
8.36 X is normally distributed with mean 1,000 and stan-
dard deviation 250. What is the probability that
X lies between 800 and 1,100?
8.37 X is normally distributed with mean 50 and standard
deviation 8. What value of X is such that only 8% of
values are below it?
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CONTINUOUS PROBABILITY DISTRIBUTIONS
8.38 The long-distance calls made by the employees of a
company are normally distributed with a mean of
6.3 minutes and a standard deviation of 2.2 minutes.
Find the probability that a call
a. lasts between 5 and 10 minutes.
b. lasts more than 7 minutes.
c. lasts less than 4 minutes.
8.39 Refer to Exercise 8.38. How long do the longest
10% of calls last?
8.40 The lifetimes of lightbulbs that are advertised to last
for 5,000 hours are normally distributed with a mean
of 5,100 hours and a standard deviation of 200 hours.
What is the probability that a bulb lasts longer than
the advertised figure?
8.41 Refer to Exercise 8.40. If we wanted to be sure that
285
there is a problem. There may be a problem when a
child is in the top or bottom 5% of heights. Determine the heights of 2-year-old children that could be
a problem.
8.47 Refer to Exercise 8.46. Find the probability of these
events.
a. A 2-year-old child is taller than 36 inches.
b. A 2-year-old child is shorter than 34 inches.
c. A 2-year-old child is between 30 and 33 inches
tall.
8.48 University and college students average 7.2 hours of
sleep per night, with a standard deviation of 40 minutes. If the amount of sleep is normally distributed,
what proportion of university and college students
sleep for more than 8 hours?
98% of all bulbs last longer than the advertised figure, what figure should be advertised?
8.49 Refer to Exercise 8.48. Find the amount of sleep that
8.42 Travelbyus is an Internet-based travel agency wherein
8.50 The amount of time devoted to studying statistics
customers can see videos of the cities they plan to
visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a
standard deviation of 2,400.
a. What is the probability of getting more than
12,000 hits?
b. What is the probability of getting fewer than
9,000 hits?
8.43 Refer to Exercise 8.42. Some Internet sites have band-
widths that are not sufficient to handle all their traffic,
often causing their systems to crash. Bandwidth can be
measured by the number of hits a system can handle.
How large a bandwidth should Travelbyus have in
order to handle 99.9% of daily traffic?
is exceeded by only 25% of students.
each week by students who achieve a grade of A in
the course is a normally distributed random variable
with a mean of 7.5 hours and a standard deviation of
2.1 hours.
a. What proportion of A students study for more
than 10 hours per week?
b. Find the probability that an A student spends
between 7 and 9 hours studying.
c. What proportion of A students spend fewer than
3 hours studying?
d. What is the amount of time below which only
5% of all A students spend studying?
8.51 The number of pages printed before replacing the
market. The distance traveled on 1 gallon of fuel is
normally distributed with a mean of 65 miles and a
standard deviation of 4 miles. Find the probability of
the following events.
a. The car travels more than 70 miles per gallon.
b. The car travels less than 60 miles per gallon.
c. The car travels between 55 and 70 miles per gallon.
cartridge in a laser printer is normally distributed
with a mean of 11,500 pages and a standard deviation of 800 pages. A new cartridge has just been
installed.
a. What is the probability that the printer produces
more than 12,000 pages before this cartridge
must be replaced?
b. What is the probability that the printer produces
fewer than 10,000 pages?
8.45 The top-selling Red and Voss tire is rated 70,000
8.52 Refer to Exercise 8.51. The manufacturer wants to
miles, which means nothing. In fact, the distance the
tires can run until they wear out is a normally distributed random variable with a mean of 82,000
miles and a standard deviation of 6,400 miles.
a. What is the probability that a tire wears out
before 70,000 miles?
b. What is the probability that a tire lasts more than
100,000 miles?
provide guidelines to potential customers advising
them of the minimum number of pages they can
expect from each cartridge. How many pages should
it advertise if the company wants to be correct 99%
of the time?
8.44 A new gas–electric hybrid car has recently hit the
8.46 The heights of children 2 years old are normally dis-
tributed with a mean of 32 inches and a standard
deviation of 1.5 inches. Pediatricians regularly measure the heights of toddlers to determine whether
8.53 Battery manufacturers compete on the basis of the
amount of time their products last in cameras and
toys. A manufacturer of alkaline batteries has
observed that its batteries last for an average of
26 hours when used in a toy racing car. The amount
of time is normally distributed with a standard deviation of 2.5 hours.
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a. What is the probability that the battery lasts
between 24 and 28 hours?
b. What is the probability that the battery lasts
longer than 28 hours?
c. What is the probability that the battery lasts less
than 24 hours?
8.54 Because of the relatively high interest rates, most
consumers attempt to pay off their credit card bills
promptly. However, this is not always possible. An
analysis of the amount of interest paid monthly by a
bank’s Visa cardholders reveals that the amount is
normally distributed with a mean of $27 and a standard deviation of $7.
a. What proportion of the bank’s Visa cardholders
pay more than $30 in interest?
b. What proportion of the bank’s Visa cardholders
pay more than $40 in interest?
c. What proportion of the bank’s Visa cardholders
pay less than $15 in interest?
d. What interest payment is exceeded by only 20%
of the bank’s Visa cardholders?
8.55 It is said that sufferers of a cold virus experience
symptoms for 7 days. However, the amount of time
is actually a normally distributed random variable
whose mean is 7.5 days and whose standard deviation is 1.2 days.
a. What proportion of cold sufferers experience
fewer than 4 days of symptoms?
b. What proportion of cold sufferers experience
symptoms for between 7 and 10 days?
8.56 How much money does a typical family of four
spend at a McDonald’s restaurant per visit? The
amount is a normally distributed random variable
with a mean of $16.40 and a standard deviation of
$2.75.
a. Find the probability that a family of four spends
less than $10.
b. What is the amount below which only 10% of
families of four spend at McDonald’s?
8.57 The final marks in a statistics course are normally
distributed with a mean of 70 and a standard deviation of 10. The professor must convert all marks to
letter grades. She decides that she wants 10% A’s,
30% B’s, 40% C’s, 15% D’s, and 5% F’s. Determine
the cutoffs for each letter grade.
8.58 Mensa is an organization whose members possess
IQs that are in the top 2% of the population. It is
known that IQs are normally distributed with a
mean of 100 and a standard deviation of 16. Find the
minimum IQ needed to be a Mensa member.
8.59 According to the 2001 Canadian census, university-
educated Canadians earned a mean income of $61,823.
The standard deviation is $17,301. If incomes are normally distributed, what is the probability that a
randomly selected university-educated Canadian earns
more than $70,000?
8.60 The census referred to in the previous exercise also
reported that college-educated Canadians earn on
average $41,825. Suppose that incomes are normally
distributed with a standard deviation of $13,444.
Find the probability that a randomly selected college-educated Canadian earns less than $45,000.
8.61 The lifetimes of televisions produced by the Hishobi
Company are normally distributed with a mean of
75 months and a standard deviation of 8 months. If
the manufacturer wants to have to replace only 1%
of its televisions, what should its warranty be?
8.62 According to the Statistical Abstract of the United
States, 2000 (Table 764), the mean family net worth
of families whose head is between 35 and 44 years
old is approximately $99,700. If family net worth is
normally distributed with a standard deviation of
$30,000, find the probability that a randomly
selected family whose head is between 35 and
44 years old has a net worth greater than $150,000.
8.63 A retailer of computing products sells a variety of
computer-related products. One of his most popular
products is an HP laser printer. The average weekly
demand is 200. Lead time for a new order from the
manufacturer to arrive is 1 week. If the demand for
printers were constant, the retailer would reorder
when there were exactly 200 printers in inventory.
However, the demand is a random variable. An
analysis of previous weeks reveals that the weekly
demand standard deviation is 30. The retailer knows
that if a customer wants to buy an HP laser printer
but he has none available, he will lose that sale plus
possibly additional sales. He wants the probability of
running short in any week to be no more than 6%.
How many HP laser printers should he have in stock
when he reorders from the manufacturer?
8.64 The demand for a daily newspaper at a newsstand at a
busy intersection is known to be normally distributed
with a mean of 150 and a standard deviation of 25.
How many newspapers should the newsstand operator order to ensure that he runs short on no more
than 20% of days?
8.65 Every day a bakery prepares its famous marble rye. A
statistically savvy customer determined that daily
demand is normally distributed with a mean of 850
and a standard deviation of 90. How many loaves
should the bakery make if it wants the probability of
running short on any day to be no more than 30%?
8.66 Refer to Exercise 8.65. Any marble ryes that are
unsold at the end of the day are marked down and
sold for half-price. How many loaves should the
bakery prepare so that the proportion of days that
result in unsold loaves is no more than 60%?
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287
APPLICATIONS in OPERATIONS MANAGEMENT
© Banana Stock/Jupiterimages
PERT/CPM
In the Applications in Operations Management box on page 235, we introduced
PERT/CPM. The purpose of this powerful management-science procedure is to
determine the critical path of a project. The expected value and variance of the
completion time of the project are based on the expected values and variances
of the completion times of the activities on the critical path. Once we have the
expected value and variance of the completion time of the project, we can use
these figures to determine the probability that the project will be completed by a
certain date. Statisticians have established that the completion time of the project is
approximately normally distributed, enabling us to compute the needed probabilities.
8.67 Refer to Exercise 7.57. Find the probability that the project will take more than
60 days to complete.
8.68 The mean and variance of the time to complete the project in Exercise 7.58 was
145 minutes and 31 minutes2. What is the probability that it will take less than
2.5 hours to overhaul the machine?
8.69 The annual rate of return on a mutual fund is nor-
8.70 In Exercise 7.64, we discovered that the expected
mally distributed with a mean of 14% and a standard
deviation of 18%.
a. What is the probability that the fund returns
more than 25% next year?
b. What is the probability that the fund loses money
next year?
return is .1060 and the standard deviation is .1456.
Working with the assumption that returns are normally distributed, determine the probability of the
following events.
a. The portfolio loses money.
b. The return on the portfolio is greater than 20%.
8 . 3 (O P T I O N A L ) E X P O N E N T I A L D I S T R I B U T I O N
Another important continuous distribution is the exponential distribution.
Exponential Probability Density Function
A random variable X is exponentially distributed if its probability density
function is given by
f1x2 = le - lx, x Ú 0
where e ⫽ 2.71828 . . . and is the parameter of the distribution.
Statisticians have shown that the mean and standard deviation of an exponential
random variable are equal to each other:
m = s = 1>l
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Recall that the normal distribution is a two-parameter distribution. The distribution is completely specified once the values of the two parameters and are known.
In contrast, the exponential distribution is a one-parameter distribution. The distribution is completely specified once the value of the parameter is known. Figure 8.21
depicts three exponential distributions, corresponding to three different values of the
parameter . Notice that for any exponential density function f1x2, f102 = l and f1x2
approaches 0 as x approaches infinity.
FIGURE
8.21
Exponential Distributions
f(x)
2
l=2
1
l=1
.5
l = .5
x
The exponential density function is easier to work with than the normal. As a
result, we can develop formulas for the calculation of the probability of any range of
values. Using integral calculus, we can determine the following probability statements.
Probability Associated with an Exponential Random Variable
If X is an exponential random variable,
P1X 7 x2 = e - lx
P1X 6 x2 = 1 - e - lx
P1x1 6 X 6 x22 = P1X 6 x22 - P1X 6 x12 = e - lx1 - e - lx2
The value of e⫺ ␭x can be obtained with the aid of a calculator.
EXAMPLE
8.7
Lifetimes of Alkaline Batteries
The lifetime of an alkaline battery (measured in hours) is exponentially distributed with
⫽ .05.
a. What is the mean and standard deviation of the battery’s lifetime?
b. Find the probability that a battery will last between 10 and 15 hours.
c. What is the probability that a battery will last for more than 20 hours?
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CONTINUOUS PROBABILITY DISTRIBUTIONS
289
SOLUTION
a. The mean and standard deviation are equal to 1/␭. Thus,
m = s = 1>l = 1>.05 = 20 hours
b. Let X denote the lifetime of a battery. The required probability is
P110 6 X 6 152 = e - .051102 - e - .051152
= e - .5 - e - .75
= .6065 - .4724
= .1341
c. P1X 7 20) = e - .051202
= e-1
= .3679
Figure 8.22 depicts these probabilities.
8.22
FIGURE
Probabilities for Example 8.7
.06
.06
.05
.05
.04
.04
P(10 < X < 15) = .1341
.03
.03
.02
.02
.01
.01
0
10
20
30
40
50
60
(b) P(10 < X < 15)
70
80
P(X > 20) = .3679
0
10
20
30
40
50
60
70
80
(c) P(X > 20)
Using the Computer
EXCEL
INSTRUCTIONS
Type (in any cell)
= EXPONDIST 13X4,3l4, True2
To produce the answer for Example 8.7c we would find P1X 6 202 and subtract it from 1.
To find P1X 6 202, type
= EXPONDIST120, .05, True2
which outputs .6321 and hence P1X 7 202 = 1 - .6321 = .3679, which is exactly the
number we produced manually.
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MINITAB
INSTRUCTIONS
Click Calc, Probability Distributions, and Exponential . . . and specify Cumulative
probability. In the Scale box, type the mean, which is 1/. In the Threshold box, type 0.
APPLICATIONS in OPERATIONS MANAGEMENT
Waiting Lines
In Section 7.5, we described waiting-line models and how the Poisson distribution is
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Getty Images
used to calculate the probabilities of the number of arrivals per time period. To calculate the operating characteristics of waiting lines, management scientists often
assume that the times to complete a service are exponentially distributed. In this
application, the parameter is the service rate, which is defined as the mean number of service completions per time period. For example, if service times are exponentially distributed with ⫽ 5/hour, this tells us that the service rate is 5 units per hour or
5 per 60 minutes. Recall that the mean of an exponential distribution is ⫽ 1/. In this case,
the service facility can complete a service in an average of 12 minutes. This was calculated as
m =
1
1
60 minutes
1
=
=
=
= 12 minutes.
l
5>hr,
5>60 minutes
5
We can use this distribution to make a variety of probability statements.
EXAMPLE
8.8
Supermarket Checkout Counter
A checkout counter at a supermarket completes the process according to an exponential
distribution with a service rate of 6 per hour. A customer arrives at the checkout
counter. Find the probability of the following events.
a. The service is completed in fewer than 5 minutes
b. The customer leaves the checkout counter more than 10 minutes after arriving
c. The service is completed in a time between 5 and 8 minutes
SOLUTION
One way to solve this problem is to convert the service rate so that the time period is
1 minute. (Alternatively, we can solve by converting the probability statements so that the
time periods are measured in fractions of an hour.) Let the service rate ⫽ ⫽ .1/minute.
a. P1X 6 52 = 1 - e - lx = 1 - e - .1152 = 1 - e - .5 = 1 - .6065 = .3935
b. P1X 7 102 = e - lx = e - .11102 = e - 1 = .3679
c. P15 6 X 6 82 = e - .1152 - e - .1182 = e - .5 - e - .8 = .6065 - .4493 = .1572
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EXERCISES
8.71 The random variable X is exponentially distributed
with ⫽ 3. Sketch the graph of the distribution of X
by plotting and connecting the points representing
f1x2 for x ⫽ 0, .5, 1, 1.5, and 2.
8.72 X is an exponential random variable with ⫽ .25.
Sketch the graph of the distribution of X by plotting
and connecting the points representing f1x2 for
x ⫽ 0, 2, 4, 6, 8, 10, 15, 20.
8.73 Let X be an exponential random variable with
␭ ⫽ .5. Find the following probabilities.
a. P1X 7 12
b. P1X 7 .42
c. P1X 6 .52
d. P1X 6 22
8.74 X is an exponential random variable with ⫽ .3.
Find the following probabilities.
a. P1X 7 22
b. P1X 6 42
c. P11 6 X 6 22
d. P1X = 32
8.75 The production of a complex chemical needed for
anticancer drugs is exponentially distributed with
⫽ 6 kilograms per hour. What is the probability
that the production process requires more than
15 minutes to produce the next kilogram of drugs?
8.76 The time between breakdowns of aging machines is
known to be exponentially distributed with a mean
of 25 hours. The machine has just been repaired.
Determine the probability that the next breakdown
occurs more than 50 hours from now.
8.77 When trucks arrive at the Ambassador Bridge, each
truck must be checked by customs agents. The times
are exponentially distributed with a service rate of
10 per hour. What is the probability that a truck
requires more than 15 minutes to be checked?
8.78 A bank wishing to increase its customer base adver-
tises that it has the fastest service and that virtually all
of its customers are served in less than 10 minutes. A
management scientist has studied the service times
and concluded that service times are exponentially
distributed with a mean of 5 minutes. Determine
what the bank means when it claims “virtually all” its
customers are served in under 10 minutes.
8.79 Toll booths on the New York State Thruway are
often congested because of the large number of cars
waiting to pay. A consultant working for the state
concluded that if service times are measured from
the time a car stops in line until it leaves, service
times are exponentially distributed with a mean of
2.7 minutes. What proportion of cars can get
through the toll booth in less than 3 minutes?
8.80 The manager of a gas station has observed that the
times required by drivers to fill their car’s tank and
pay are quite variable. In fact, the times are exponentially distributed with a mean of 7.5 minutes.
What is the probability that a car can complete the
transaction in less than 5 minutes?
8.81 Because automatic banking machine (ABM) cus-
tomers can perform a number of transactions, the
times to complete them can be quite variable. A
banking consultant has noted that the times are
exponentially distributed with a mean of 125 seconds. What proportion of the ABM customers take
more than 3 minutes to do their banking?
8.82 The manager of a supermarket tracked the amount
of time needed for customers to be served by the
cashier. After checking with his statistics professor,
he concluded that the checkout times are exponentially distributed with a mean of 6 minutes. What
proportion of customers require more than 10 minutes to check out?
8 . 4 OT H E R CON T I N U O U S DI S T R I B U T ION S
In this section, we introduce three more continuous distributions that are used extensively in statistical inference.
Student t Distribution
The Student t distribution was first derived by William S. Gosset in 1908. (Gosset
published his findings under the pseudonym “Student” and used the letter t to represent the random variable, hence the Student t distribution—also called the Student’s t
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distribution.) It is very commonly used in statistical inference, and we will employ it in
Chapters 12, 13, 14, 16, 17, and 18.
Student t Density Function
The density function of the Student t distribution is as follows:
f1t2 =
≠31n + 12>24
2np≠1n>22
B1 +
t2 - 1n + 12>2
R
n
where ␯ (Greek letter nu) is the parameter of the Student t distribution
called the degrees of freedom, ⫽ 3.14159 (approximately), and ⌫ is the
gamma function (its definition is not needed here).
The mean and variance of a Student t random variable are
E1t2 = 0
and
V1t2 =
n
for n 7 2
n - 2
Figure 8.23 depicts the Student t distribution. As you can see, it is similar to the
standard normal distribution. Both are symmetrical about 0. (Both random variables
have a mean of 0.) We describe the Student t distribution as mound shaped, whereas the
normal distribution is bell shaped.
FIGURE
8.23
Student t Distribution
t
0
Figure 8.24 shows both a Student t and the standard normal distributions. The former is more widely spread out than the latter. [The variance of a standard normal random variable is 1, whereas the variance of a Student t random variable is n>1n - 22,
which is greater than 1 for all ␯.]
FIGURE
8.24
Student t and Normal Distributions
Standard normal
distribution
Student t distribution
0
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CONTINUOUS PROBABILITY DISTRIBUTIONS
293
Figure 8.25 depicts Student t distributions with several different degrees of freedom. Notice that for larger degrees of freedom the Student t distribution’s dispersion is
smaller. For example, when ␯ ⫽ 10, V(t)⫽1.25; when ␯ ⫽ 50, V(t)⫽1.042; and when
␯ ⫽ 200, V(t)⫽1.010. As ␯ grows larger, the Student t distribution approaches the standard normal distribution.
FIGURE
8.25
Student t Distribution with ␯ = 2, 10, and 30
Student t
with n = 10
Student t
with n = 30
Student t
with n = 2
0
Student t Probabilities For each value of ␯ (the number of degrees of freedom),
there is a different Student t distribution. If we wanted to calculate probabilities of
the Student t random variable manually as we did for the normal random variable,
then we would need a different table for each ␯, which is not practical. Alternatively,
we can use Microsoft Excel or Minitab. The instructions are given later in this
section.
Determining Student t Values As you will discover later in this book, the Student
t distribution is used extensively in statistical inference. And for inferential methods, we
often need to find values of the random variable. To determine values of a normal random variable, we used Table 3 backward. Finding values of a Student t random variable
is considerably easier. Table 4 in Appendix B (reproduced here as Table 8.2) lists values
of tA,n, which are the values of a Student t random variable with ␯ degrees of freedom
such that
P1t 7 tA,n2 = A
Figure 8.26 depicts this notation.
FIGURE
8.26
Student t Distribution with t A
A
t
0
tA
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TABLE
8. 2
␯
Critical Values of t
t.100
t.050
t.025
t.010
t.005
12.71
31.82
63.66
␯
t.100
t.050
t.025
t.010
t.005
1
3.078
6.314
29
1.311
1.699
2.045
2.462
2.756
2
1.886
2.920
4.303
6.965
9.925
30
1.310
1.697
2.042
2.457
2.750
3
1.638
2.353
3.182
4.541
5.841
35
1.306
1.690
2.030
2.438
2.724
4
1.533
2.132
2.776
3.747
4.604
40
1.303
1.684
2.021
2.423
2.704
5
1.476
2.015
2.571
3.365
4.032
45
1.301
1.679
2.014
2.412
2.690
6
1.440
1.943
2.447
3.143
3.707
50
1.299
1.676
2.009
2.403
2.678
7
1.415
1.895
2.365
2.998
3.499
55
1.297
1.673
2.004
2.396
2.668
8
1.397
1.860
2.306
2.896
3.355
60
1.296
1.671
2.000
2.390
2.660
9
1.383
1.833
2.262
2.821
3.250
65
1.295
1.669
1.997
2.385
2.654
10
1.372
1.812
2.228
2.764
3.169
70
1.294
1.667
1.994
2.381
2.648
11
1.363
1.796
2.201
2.718
3.106
75
1.293
1.665
1.992
2.377
2.643
12
1.356
1.782
2.179
2.681
3.055
80
1.292
1.664
1.990
2.374
2.639
13
1.350
1.771
2.160
2.650
3.012
85
1.292
1.663
1.988
2.371
2.635
14
1.345
1.761
2.145
2.624
2.977
90
1.291
1.662
1.987
2.368
2.632
15
1.341
1.753
2.131
2.602
2.947
95
1.291
1.661
1.985
2.366
2.629
16
1.337
1.746
2.120
2.583
2.921
100
1.290
1.660
1.984
2.364
2.626
17
1.333
1.740
2.110
2.567
2.898
110
1.289
1.659
1.982
2.361
2.621
18
1.330
1.734
2.101
2.552
2.878
120
1.289
1.658
1.980
2.358
2.617
19
1.328
1.729
2.093
2.539
2.861
130
1.288
1.657
1.978
2.355
2.614
20
1.325
1.725
2.086
2.528
2.845
140
1.288
1.656
1.977
2.353
2.611
21
1.323
1.721
2.080
2.518
2.831
150
1.287
1.655
1.976
2.351
2.609
22
1.321
1.717
2.074
2.508
2.819
160
1.287
1.654
1.975
2.350
2.607
23
1.319
1.714
2.069
2.500
2.807
170
1.287
1.654
1.974
2.348
2.605
24
1.318
1.711
2.064
2.492
2.797
180
1.286
1.653
1.973
2.347
2.603
25
1.316
1.708
2.060
2.485
2.787
190
1.286
1.653
1.973
2.346
2.602
26
1.315
1.706
2.056
2.479
2.779
200
1.286
1.653
1.972
2.345
2.601
⬁
1.282
1.645
1.960
2.326
2.576
27
1.314
1.703
2.052
2.473
2.771
28
1.313
1.701
2.048
2.467
2.763
Observe that tA,␯ is provided for degrees of freedom ranging from 1 to 200 and ⬁.
To read this table, simply identify the degrees of freedom and find that value or the closest number to it if it is not listed. Then locate the column representing the tA value you
wish. For example, if we want the value of t with 10 degrees of freedom such that the area
under the Student t curve is .05, we locate 10 in the first column and move across this
row until we locate the number under the heading t.05. From Table 8.3, we find
t.05,10 ⫽ 1.812
If the number of degrees of freedom is not shown, find its closest value. For example, suppose we wanted to find t.025,32. Because 32 degrees of freedom is not listed, we
find the closest number of degrees of freedom, which is 30 and use t.025,30 ⫽ 2.042 as an
approximation.
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TABLE
8. 3
295
Finding t.05,10
DEGREES OF
FREEDOM
t.10
t.05
t.025
t.01
t.005
1
3.078
6.314
12.706
31.821
63.657
2
1.886
2.920
4.303
6.965
9.925
3
1.638
2.353
3.182
4.541
5.841
4
1.533
2.132
2.776
3.747
4.604
5
1.476
2.015
2.571
3.365
4.032
6
1.440
1.943
2.447
3.143
3.707
7
1.415
1.895
2.365
2.998
3.499
8
1.397
1.860
2.306
2.896
3.355
9
1.383
1.833
2.262
2.821
3.250
10
1.372
1.812
2.228
2.764
3.169
11
1.363
1.796
2.201
2.718
3.106
12
1.356
1.782
2.179
2.681
3.055
Because the Student t distribution is symmetric about 0, the value of t such that the
area to its left is A is ⫺ tA,. For example, the value of t with 10 degrees of freedom such
that the area to its left is .05 is
- t.05,10 = - 1.812
Notice the last row in the Student t table. The number of degrees of freedom is
infinite, and the t values are identical (except for the number of decimal places) to the
values of z. For example,
t.10, q = 1.282
t.05, q = 1.645
t.025, q = 1.960
t.01, q = 2.326
t.005, q = 2.576
In the previous section, we showed (or showed how we determine) that
z.10 = 1.28
z.05 = 1.645
z.025 = 1.96
z.01 = 2.23
z.005 = 2.575
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Using the Computer
EXCEL
INSTRUCTIONS
To compute Student t probabilities, type
= TDIST13x4, 3n4, 3Tails42
where x must be positive, ␯ is the number of degrees of freedom, and “Tails” is 1 or 2.
Typing 1 for “Tails” produces the area to the right of x. Typing 2 for “Tails” produces the
area to the right of x plus the area to the left of ⫺x. For example,
= TDIST12, 50, 12 = .02547
and
= TDIST12, 50, 22 = .05095
To determine tA, type
= TINV132A4, 3n42
For example, to find t.05,200 enter
= TINV1.10, 2002
yielding 1.6525.
MINITAB
INSTRUCTIONS
Click Calc, Probability Distributions, and t . . . and type the Degrees of freedom.
S E E I N G S TAT I S T I C S
applet 6 Student t Distribution
The Student t distribution applet allows
(blue curve). Use the right slider to
you to see for yourself the shape of the
change the degrees of freedom for the
distribution, how the degrees of
t distribution. Use the text boxes to
freedom change the shape, and its
change either the value of t or the two-
resemblance to the standard normal
tail probability. Remember to press the
curve. The first graph shows the
Return key in the text box to record the
comparison of the normal distribution
change.
(red curve) to the Student t distribution
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The second graph is the same as the one
above except the comparison to the
297
6.2 Again for df ⫽ 2, for values around
either ⫹4 or ⫺4, which curve is
normal distribution has been removed.
higher? In other words, which
This graph is a little easier to use to find
distribution is more likely to have
critical values of t or to find the
extreme values—the normal (red) or
probability of specific values of t.
Student t (blue) distribution?
Applet Exercises
The following exercises refer to Graph 2.
The following exercises refer to Graph 1.
6.3 As you use the scrollbar to increase
6.1 Set the degrees of freedom equal
(slowly) the degrees of freedom,
between the critical values of t.025
to 2. For values (on the horizontal
what happens to the value of t.025
and z.025? How large do you think
axis) near 0, which curve is higher?
and ⫺t.025?
the degrees of freedom would have
The higher curve is more likely to
6.4 When the degrees of freedom ⫽
have observations in that region.
100, is there still a small difference
to be before the two sets of critical
values were identical?
Chi-Squared Distribution The density function of another very useful random variable is exhibited next.
Chi-Squared Density Function
The chi-squared density function is
f1x22 =
1
1
2
1x221n>22 - 1e - x >2 x2 7 0
n>2
≠1n>22 2
The parameter ␯ is the number of degrees of freedom, which like the
degrees of freedom of the Student t distribution affects the shape.
Figure 8.27 depicts a chi-squared distribution. As you can see, it is positively
skewed ranging between 0 and ⬁. Like that of the Student t distribution, its shape
depends on its number of degrees of freedom. The effect of increasing the degrees of
freedom is seen in Figure 8.28.
FIGURE
8.27
FIGURE 8.28 Chi-Squared Distribution with
␯ = 1, 5, and 10
Chi-Squared Distribution
f(x2)
f(x2)
n=1
0
x2
n=5
n = 10
0
x2
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CHAPTER 8
The mean and variance of a chi-squared random variable are
E1x22 = n
and
V1x22 = 2n
Determining Chi-Squared Values The value of ␹2 with ␯ degrees of freedom such
that the area to its right under the chi-squared curve is equal to A is denoted ␹2A,␯. We
cannot use ⫺␹2A,␯ to represent the point such that the area to its left is A (as we did with
the standard normal and Student t values) because ␹2 is always greater than 0. To represent left-tail critical values, we note that if the area to the left of a point is A, the area to
its right must be 1 ⫺ A because the entire area under the chi-squared curve (as well as
all continuous distributions) must equal 1. Thus, x21 - A,n denotes the point such that the
area to its left is A. See Figure 8.29.
FIGURE
8.29
␹ 2A and ␹ 21-A
f(x2)
A
A
x2
2
0 x1 – A
x2A
Table 5 in Appendix B (reproduced here as Table 8.4) lists critical values of the
chi-squared distribution for degrees of freedom equal to 1 to 30, 40, 50, 60, 70, 80, 90,
and 100. For example, to find the point in a chi-squared distribution with 8 degrees of
freedom such that the area to its right is .05, locate 8 degrees of freedom in the left column and x2.050 across the top. The intersection of the row and column contains the
number we seek as shown in Table 8.5; that is,
x2.050,8 = 15.5
To find the point in the same distribution such that the area to its left is .05, find the
point such that the area to its right is .95. Locate x2.950 across the top row and 8 degrees
of freedom down the left column (also shown in Table 8.5). You should see that
x2.950,8 = 2.73
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TABLE
8. 4
Page 299
Critical Values of ␹2
␯
x2.995
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
0.000039
0.0100
0.072
0.207
0.412
0.676
0.989
1.34
1.73
2.16
2.60
3.07
3.57
4.07
4.60
5.14
5.70
6.26
6.84
7.43
8.03
8.64
9.26
9.89
10.52
11.16
11.81
12.46
13.12
13.79
20.71
27.99
35.53
43.28
51.17
59.20
67.33
TABLE
8. 5
0.000157
0.0201
0.115
0.297
0.554
0.872
1.24
1.65
2.09
2.56
3.05
3.57
4.11
4.66
5.23
5.81
6.41
7.01
7.63
8.26
8.90
9.54
10.20
10.86
11.52
12.20
12.88
13.56
14.26
14.95
22.16
29.71
37.48
45.44
53.54
61.75
70.06
x2.975
x2.950
0.000982
0.0506
0.216
0.484
0.831
1.24
1.69
2.18
2.70
3.25
3.82
4.40
5.01
5.63
6.26
6.91
7.56
8.23
8.91
9.59
10.28
10.98
11.69
12.40
13.12
13.84
14.57
15.31
16.05
16.79
24.43
32.36
40.48
48.76
57.15
65.65
74.22
x2.900
0.00393
0.103
0.352
0.711
1.15
1.64
2.17
2.73
3.33
3.94
4.57
5.23
5.89
6.57
7.26
7.96
8.67
9.39
10.12
10.85
11.59
12.34
13.09
13.85
14.61
15.38
16.15
16.93
17.71
18.49
26.51
34.76
43.19
51.74
60.39
69.13
77.93
x2.100
0.0158
0.211
0.584
1.06
1.61
2.20
2.83
3.49
4.17
4.87
5.58
6.30
7.04
7.79
8.55
9.31
10.09
10.86
11.65
12.44
13.24
14.04
14.85
15.66
16.47
17.29
18.11
18.94
19.77
20.60
29.05
37.69
46.46
55.33
64.28
73.29
82.36
2.71
4.61
6.25
7.78
9.24
10.6
12.0
13.4
14.7
16.0
17.3
18.5
19.8
21.1
22.3
23.5
24.8
26.0
27.2
28.4
29.6
30.8
32.0
33.2
34.4
35.6
36.7
37.9
39.1
40.3
51.8
63.2
74.4
85.5
96.6
108
118
x2.050
x2.025
3.84
5.02
5.99
7.38
7.81
9.35
9.49
11.1
11.1
12.8
12.6
14.4
14.1
16.0
15.5
17.5
16.9
19.0
18.3
20.5
19.7
21.9
21.0
23.3
22.4
24.7
23.7
26.1
25.0
27.5
26.3
28.8
27.6
30.2
28.9
31.5
30.1
32.9
31.4
34.2
32.7
35.5
33.9
36.8
35.2
38.1
36.4
39.4
37.7
40.6
38.9
41.9
40.1
43.2
41.3
44.5
42.6
45.7
43.8
47.0
55.8
59.3
67.5
71.4
79.1
83.3
90.5
95.0
102
107
113
118
124
130
x2.010
x2.005
6.63
7.88
9.21
10.6
11.3
12.8
13.3
14.9
15.1
16.7
16.8
18.5
18.5
20.3
20.1
22.0
21.7
23.6
23.2
25.2
24.7
26.8
26.2
28.3
27.7
29.8
29.1
31.3
30.6
32.8
32.0
34.3
33.4
35.7
34.8
37.2
36.2
38.6
37.6
40.0
38.9
41.4
40.3
42.8
41.6
44.2
43.0
45.6
44.3
46.9
45.6
48.3
47.0
49.6
48.3
51.0
49.6
52.3
50.9
53.7
63.7
66.8
76.2
79.5
88.4
92.0
100
104
112
116
124
128
136
140
Critical Values of x2.05,8 and x2.950,8
DEGREES OF
FREEDOM
1
2
3
4
5
6
7
8
9
10
11
x2.990
x2.995
x2.990
x2.975
x2.950
x2.900
x2.100
x2.050
x2.025
x2.010
x2.005
0.000039
0.0100
0.072
0.207
0.412
0.676
0.989
1.34
1.73
2.16
2.60
0.000157
0.0201
0.115
0.297
0.554
0.872
1.24
1.65
2.09
2.56
3.05
0.000982
0.0506
0.216
0.484
0.831
1.24
1.69
2.18
2.70
3.25
3.82
0.00393
0.103
0.352
0.711
1.15
1.64
2.17
2.73
3.33
3.94
4.57
0.0158
0.211
0.584
1.06
1.61
2.20
2.83
3.49
4.17
4.87
5.58
2.71
4.61
6.25
7.78
9.24
10.6
12.0
13.4
14.7
16.0
17.3
3.84
5.99
7.81
9.49
11.1
12.6
14.1
15.5
16.9
18.3
19.7
5.02
7.38
9.35
11.1
12.8
14.4
16.0
17.5
19.0
20.5
21.9
6.63
9.21
11.3
13.3
15.1
16.8
18.5
20.1
21.7
23.2
24.7
7.88
10.6
12.8
14.9
16.7
18.5
20.3
22.0
23.6
25.2
26.8
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CHAPTER 8
For values of degrees of freedom greater than 100, the chi-squared distribution can
be approximated by a normal distribution with ␮ ⫽ ␯ and s = 22n.
Using the Computer
EXCEL
INSTRUCTIONS
To calculate P1x2 7 x2, type into any cell
= CHIDIST13x4, 3n42
For example, CHIDIST16.25, 32 = .100.
To determine xA,n, type
= CHIINV13A4, 3n42
For example, = CHIINV1.10, 32 = 6.25
MINITAB
INSTRUCTIONS
Click Calc, Probability Distributions, and Chi-square . . . . Specify the Degrees of
freedom.
S E E I N G S TAT I S T I C S
applet 7 Chi-Squared Distribution
Like the Student t applet, this applet
the probability. Remember to press the
allows you to see how the degrees of
Return key in the text box to record the
freedom affect the shape of the chi-
change.
squared distribution. Additionally, you
can use the applet to determine
Applet Exercises
probabilities and values of the chi-
7.1 What happens to the shape of the
squared random variable.
Use the right slider to change the
chi-squared distribution as the
degrees of freedom increase?
degrees of freedom. Use the text boxes
7.2 Describe what happens to x2.05 when
7.3 Describe what happens to x2.95 when
to change either the value of ChiSq or
the degrees of freedom increase.
the degrees of freedom increase.
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CONTINUOUS PROBABILITY DISTRIBUTIONS
301
F Distribution
The density function of the F distribution is given in the box.
F Density Function
≠¢
f1F2 =
≠¢
n1 + n2
2
n1
2
≤≠¢
n2
2
n1
≤
¢
≤
n1
n2
≤
F
2
¢1 +
n1 - 2
2
n1F
n2
n1 + n2
≤
F 7 0
2
where F ranges from 0 to ⬁ and ␯1 and ␯2 are the parameters of the distribution called degrees of freedom. For reasons that are clearer in Chapter 13,
we call ␯1 the numerator degrees of freedom and ␯2 the denominator degrees of
freedom.
The mean and variance of an F random variable are
E1F2 =
n2
n2 - 2
n2 7 2
and
V1F2 =
2n221n1 + n2 - 22
n11n2 - 2221n2 - 42
n2 7 4
Notice that the mean depends only on the denominator degrees of freedom and that for
large ␯2 the mean of the F distribution is approximately 1. Figure 8.30 describes the
density function when it is graphed. As you can see, the F distribution is positively
skewed. Its actual shape depends on the two numbers of degrees of freedom.
FIGURE
8.30
F Distribution
f(F)
F
0
Determining Values of F We define FA,n ,n as the value of F with ␯1 and ␯2 degrees
1 2
of freedom such that the area to its right under the curve is A; that is,
P1F 7 FA,n ,n 2 = A
1
2
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CHAPTER 8
TABLE
8.6
Critical Values of FA for A ⫽ .05
␯1
␯2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
24
26
28
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
⬁
1
2
3
4
5
6
7
8
9
161
18.5
10.1
7.71
6.61
5.99
5.59
5.32
5.12
4.96
4.84
4.75
4.67
4.60
4.54
4.49
4.45
4.41
4.38
4.35
4.30
4.26
4.23
4.20
4.17
4.12
4.08
4.06
4.03
4.00
3.98
3.96
3.95
3.94
3.92
3.91
3.90
3.89
3.89
3.84
199
19.0
9.55
6.94
5.79
5.14
4.74
4.46
4.26
4.10
3.98
3.89
3.81
3.74
3.68
3.63
3.59
3.55
3.52
3.49
3.44
3.40
3.37
3.34
3.32
3.27
3.23
3.20
3.18
3.15
3.13
3.11
3.10
3.09
3.07
3.06
3.05
3.05
3.04
3.00
216
19.2
9.28
6.59
5.41
4.76
4.35
4.07
3.86
3.71
3.59
3.49
3.41
3.34
3.29
3.24
3.20
3.16
3.13
3.10
3.05
3.01
2.98
2.95
2.92
2.87
2.84
2.81
2.79
2.76
2.74
2.72
2.71
2.70
2.68
2.67
2.66
2.65
2.65
2.61
225
19.2
9.12
6.39
5.19
4.53
4.12
3.84
3.63
3.48
3.36
3.26
3.18
3.11
3.06
3.01
2.96
2.93
2.90
2.87
2.82
2.78
2.74
2.71
2.69
2.64
2.61
2.58
2.56
2.53
2.50
2.49
2.47
2.46
2.45
2.44
2.43
2.42
2.42
2.37
230
19.3
9.01
6.26
5.05
4.39
3.97
3.69
3.48
3.33
3.20
3.11
3.03
2.96
2.90
2.85
2.81
2.77
2.74
2.71
2.66
2.62
2.59
2.56
2.53
2.49
2.45
2.42
2.40
2.37
2.35
2.33
2.32
2.31
2.29
2.28
2.27
2.26
2.26
2.21
234
19.3
8.94
6.16
4.95
4.28
3.87
3.58
3.37
3.22
3.09
3.00
2.92
2.85
2.79
2.74
2.70
2.66
2.63
2.60
2.55
2.51
2.47
2.45
2.42
2.37
2.34
2.31
2.29
2.25
2.23
2.21
2.20
2.19
2.18
2.16
2.16
2.15
2.14
2.10
237
19.4
8.89
6.09
4.88
4.21
3.79
3.50
3.29
3.14
3.01
2.91
2.83
2.76
2.71
2.66
2.61
2.58
2.54
2.51
2.46
2.42
2.39
2.36
2.33
2.29
2.25
2.22
2.20
2.17
2.14
2.13
2.11
2.10
2.09
2.08
2.07
2.06
2.06
2.01
239
19.4
8.85
6.04
4.82
4.15
3.73
3.44
3.23
3.07
2.95
2.85
2.77
2.70
2.64
2.59
2.55
2.51
2.48
2.45
2.40
2.36
2.32
2.29
2.27
2.22
2.18
2.15
2.13
2.10
2.07
2.06
2.04
2.03
2.02
2.01
2.00
1.99
1.98
1.94
241
19.4
8.81
6.00
4.77
4.10
3.68
3.39
3.18
3.02
2.90
2.80
2.71
2.65
2.59
2.54
2.49
2.46
2.42
2.39
2.34
2.30
2.27
2.24
2.21
2.16
2.12
2.10
2.07
2.04
2.02
2.00
1.99
1.97
1.96
1.95
1.94
1.93
1.93
1.88
10
242
19.4
8.79
5.96
4.74
4.06
3.64
3.35
3.14
2.98
2.85
2.75
2.67
2.60
2.54
2.49
2.45
2.41
2.38
2.35
2.30
2.25
2.22
2.19
2.16
2.11
2.08
2.05
2.03
1.99
1.97
1.95
1.94
1.93
1.91
1.90
1.89
1.88
1.88
1.83
Because the F random variable like the chi-squared can equal only positive values,
we define F1 - A,n ,n as the value such that the area to its left is A. Figure 8.31 depicts this
1
2
notation. Table 6 in Appendix B provides values of FA,n ,n for A ⫽ .05, .025, .01, and .005.
1 2
Part of Table 6 is reproduced here as Table 8.6.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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303
CONTINUOUS PROBABILITY DISTRIBUTIONS
8.31
FIGURE
F 1– A and F A
f(F)
A
A
0
F
FA
F1 – A
Values of F1 - A,n ,n are unavailable. However, we do not need them because we can
1
2
determine F1 - A,n ,n from FA,n ,n . Statisticians can show that
1
F1 - A,n ,n =
1
2
2
1
2
1
FA,n ,n
2
1
To determine any critical value, find the numerator degrees of freedom ␯1 across
the top of Table 6 and the denominator degrees of freedom ␯2 down the left column.
The intersection of the row and column contains the number we seek. To illustrate,
suppose that we want to find F.05,5,7. Table 8.7 shows how this point is found. Locate
the numerator degrees of freedom, 5, across the top and the denominator degrees of
freedom, 7, down the left column. The intersection is 3.97. Thus, F.05,5,7 ⫽ 3.97.
TABLE
8.7
F.05,5,7
␯1
NUMERATOR DEGREES OF FREEDOM
␯2
Denominator Degrees
of Freedom
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
161
199
216
225
230
18.5
19.0
19.2
19.2
19.3
10.1
9.55
9.28
9.12
9.01
7.71
6.94
6.59
6.39
6.26
6.61
5.79
5.41
5.19
5.05
5.99
5.14
4.76
4.53
4.39
5.59
4.74
4.35
4.12
3.97
5.32
4.46
4.07
3.84
3.69
5.12
4.26
3.86
3.63
3.48
4.96
4.10
3.71
3.48
3.33
6
7
234
19.3
8.94
6.16
4.95
4.28
3.87
3.58
3.37
3.22
237
19.4
8.89
6.09
4.88
4.21
3.79
3.5
3.29
3.14
8
9
239
241
19.4
19.4
8.85
8.81
6.04
6.00
4.82
4.77
4.15
4.1
3.73
3.68
3.44
3.39
3.23
3.18
3.07
3.02
Note that the order in which the degrees of freedom appear is important. To find
F.05,7,5 (numerator degrees of freedom ⫽ 7, and denominator degrees of freedom ⫽ 5),
we locate 7 across the top and 5 down the side. The intersection is F.05,7,5 ⫽ 4.88.
Suppose that we want to determine the point in an F distribution with ␯1 ⫽ 4 and
␯2 ⫽ 8 such that the area to its right is .95. Thus,
F.95,4,8 =
1
F.05,8,4
=
1
= .166
6.04
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Using the Computer
EXCEL
INSTRUCTIONS
For probabilities, type
= FDIST13X4, 3n14, 3n242
For example, = FDIST13.97, 5, 72 = .05.
To determine FA,n ,n , type
1
2
= FINV13A4, 3n14, 3n242
For example, = FINV1.05, 5, 72 = 3.97.
MINITAB
INSTRUCTIONS
Click Calc, Probability Distributions, and F . . . . Specify the Numerator degrees of
freedom and the Denominator degrees of freedom.
S E E I N G S TAT I S T I C S
applet 8 F Distribution
The graph shows the F distribution. Use
the left and right sliders to change the
numerator and denominator degrees of
the denominator degrees of
freedom increase?
8.2 Set the numerator degrees of
freedom, respectively. Use the text boxes
freedom equal to 10. What happens
to change either the value of F or the
to the shape of the F distribution as
probability. Remember to press the Return
the denominator degrees of
key in the text box to record the change.
Applet Exercises
8.1 Set the numerator degrees of
freedom equal to 1. What happens
to the shape of the F distribution as
freedom increase?
8.3 Describe what happens to F.05 when
either the numerator or the
8.4 Describe what happens to F.95 when
denominator degrees of freedom
either the numerator or the denom-
increase.
inator degrees of freedom increase.
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CONTINUOUS PROBABILITY DISTRIBUTIONS
305
EXERCISES
Some of the following exercises require the use of a computer and
software.
8.83 Use the t table (Table 4) to find the following values
of t.
a. t.10,15
b. t.10,23
c. t.025,83
d. t.05,195
8.84 Use the t table (Table 4) to find the following values
of t.
a. t.005,33
b. t.10,600
c. t.05,4
d. t.01,20
8.85 Use a computer to find the following values of t.
a. t.10,15
b. t.10,23
c. t.025,83
d. t.05,195
8.86 Use a computer to find the following values of t.
a. t.05,143
b. t.01,12
c. t.025,⬁
d. t.05,100
8.87 Use a computer to find the following probabilities.
a. P1t64 7 2.122
c. P1t159 7 1.332
b. P1t27 7 1.902
d. P1t550 7 1.852
8.88 Use a computer to find the following probabilities.
a. P1t141 7 .942
b. P1t421 7 2.002
c. P1t1000 7 1.962
d. P1t82 7 1.962
8.89 Use the ␹2 table (Table 5) to find the following
values of ␹2.
a. x2.10,5
b. x2.01,100
c. x2.95,18
d. x2.99,60
8.90 Use the ␹2 table (Table 5) to find the following val-
ues of ␹2.
a. x2.90,26
b. x2.01,30
c. x2.10,1
d. x2.99,80
8.91 Use a computer to find the following values of ␹2.
a.
x2.25,66
b.
x2.40,100
c.
x2.50,17
d.
x2.10,17
8.92 Use a computer to find the following values of ␹2.
a. x2.99,55
b. x2.05,800
c. x2.99,43
d. x2.10,233
8.93 Use a computer to find the following probabilities.
a. P1x273 7 802
c. P1x288 7 602
b. P1x2200 7 1252
d. P1x21000 7 4502
8.94 Use a computer to find the following probabilities.
a. P1x2250 7 2502
c. P1x2600 7 5002
b. P1x236 7 252
d. P1x2120 7 1002
8.95 Use the F table (Table 6) to find the following values
of F.
a. F.05,3,7
b. F.05,7,3
c. F.025,5,20
d. F.01,12,60
8.96 Use the F table (Table 6) to find the following values
of F.
a. F.025,8,22
c. F.01,9,18
b. F.05,20,30
d. F.025,24,10
8.97 Use a computer to find the following values of F.
a. F.05,70,70
c. F.025,36,50
b. F.01,45,100
d. F.05,500,500
8.98 Use a computer to find the following values of F.
a. F.01,100,150
c. F.01,11,33
b. F.05,25,125
d. F.05,300,800
8.99 Use a computer to find the following probabilities.
a. P1F7,20 7 2.52
b. P1F18,63 7 1.42
c. P1F34,62 7 1.82
d. P1F200,400 7 1.12
8.100 Use a computer to find the following probabilities.
a. P1F600,800 7 1.12
b. P1F35,100 7 1.32
c. P1F66,148 7 2.12
d. P1F17,37 7 2.82
CHAPTER SUM M ARY
This chapter dealt with continuous random variables and
their distributions. Because a continuous random variable
can assume an infinite number of values, the probability
that the random variable equals any single value is 0.
Consequently, we address the problem of computing the
probability of a range of values. We showed that the probability of any interval is the area in the interval under the
curve representing the density function.
We introduced the most important distribution in
statistics and showed how to compute the probability that
a normal random variable falls into any interval.
Additionally, we demonstrated how to use the normal
table backward to find values of a normal random variable
given a probability. Next we introduced the exponential
distribution, a distribution that is particularly useful in
several management-science applications. Finally, we presented three more continuous random variables and their
probability density functions. The Student t, chisquared, and F distributions will be used extensively in
statistical inference.
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IMPORTANT TERMS
Exponential distribution 287
Student t distribution 291
Degrees of freedom 292
Chi-squared distribution 297
F distribution 301
Probability density function 265
Uniform probability distribution 266
Rectangular probability distribution 266
Normal distribution 270
Normal random variable 270
Standard normal random variable 272
SYMBOLS
Symbol
Pronounced
Represents
zA
␯
tA
x2A
pi
z-sub-A or z-A
nu
t-sub-A or t-A
chi-squared-sub-A
or chi-squared-A
F-sub-A or F-A
3.14159 . . .
Value of Z such that area to its right is A
Degrees of freedom
Value of t such that area to its right is A
Value of chi-squared such that area to its right
is A
Value of F such that area to its right is A
nu-sub-one or nu-one
nu-sub-two or nu-two
Numerator degrees of freedom
Denominator degrees of freedom
FA
␯1
␯2
COMPUTER OUTPUT AND INSTRUCTIONS
Probability/Random Variable
Normal probability
Normal random variable
Exponential probability
Exponential random variable
Student t probability
Student t random variable
Chi-squared probability
Chi-squared random variable
F probability
F random variable
Excel
Minitab
282
282
289
289
296
296
300
300
304
304
282
282
290
290
296
296
300
300
304
304
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9
© Lester Lefkowitz/Getty Images
SAMPLING DISTRIBUTIONS
9.1
Sampling Distribution of the Mean
9.2
Sampling Distribution of a Proportion
9.3
Sampling Distribution of the Difference between Two Means
9.4
From Here to Inference
Deans and other faculty members in professional schools often monitor how well the graduates of
their programs fare in the job market. Information about the types of jobs and their salaries may
provide useful information about the success of a program.
In the advertisements for a large university, the dean of the School of Business claims that the
average salary of the school’s graduates one year after graduation is $800 per week, with a standard
© Leland Bobbe/Getty Images
Salaries of a Business School’s Graduates
deviation of $100. A second-year student in the business school who has just completed his statistics
course would like to check whether the claim about the mean is correct. He does a survey of 25 people
who graduated one year earlier and determines their weekly salary. He discovers the sample mean to be
See page 317 for the
answer.
$750. To interpret his finding, he needs to calculate the probability that a sample of 25 graduates
would have a mean of $750 or less when the population mean is $800 and the standard deviation is
$100. After calculating the probability, he needs to draw some conclusion.
307
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CHAPTER 9
I N T RO D U C T I O N
T
his chapter introduces the sampling distribution, a fundamental element in statistical inference. We remind you that statistical inference is the process of converting
data into information. Here are the parts of the process we have thus far discussed:
1. Parameters describe populations.
2. Parameters are almost always unknown.
3. We take a random sample of a population to obtain the necessary data.
4. We calculate one or more statistics from the data.
For example, to estimate a population mean, we compute the sample mean.
Although there is very little chance that the sample mean and the population mean are
identical, we would expect them to be quite close. However, for the purposes of statistical inference, we need to be able to measure how close. The sampling distribution provides this service. It plays a crucial role in the process because the measure of proximity
it provides is the key to statistical inference.
9. 1
S A M PL I N G D I S T R I B U T I O N
OF THE
MEAN
A sampling distribution is created by, as the name suggests, sampling. There are two
ways to create a sampling distribution. The first is to actually draw samples of the same
size from a population, calculate the statistic of interest, and then use descriptive techniques to learn more about the sampling distribution. The second method relies on the
rules of probability and the laws of expected value and variance to derive the sampling
distribution. We’ll demonstrate the latter approach by developing the sampling distribution of the mean of two dice.
Sampling Distribution of the Mean of Two Dice
The population is created by throwing a fair die infinitely many times, with the random
variable X indicating the number of spots showing on any one throw. The probability
distribution of the random variable X is as follows:
x
p(x)
1
2
3
4
5
6
1/6
1/6
1/6
1/6
1/6
1/6
The population is infinitely large because we can throw the die infinitely many
times (or at least imagine doing so). From the definitions of expected value and variance
presented in Section 7.1, we calculate the population mean, variance, and standard
deviation.
Population mean:
m = a xP(x)
= 1(1>6) + 2(1>6) + 3(1>6) + 4(1>6) + 5(1>6) + 6(1>6)
= 3.5
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SAMPLING DISTRIBUTIONS
309
Population variance:
s2 = a (x - m)2P(x)
= (1 - 3.5)2(1>6) + (2 - 3.5)2(1>6) + (3 - 3.5)2(1>6) + (4 - 3.5)2(1>6)
+ (5 - 3.5)2(1>6) + (6 - 3.5)2(1>6)
= 2.92
Population standard deviation:
s = 2s2 = 22.92 = 1.71
The sampling distribution is created by drawing samples of size 2 from the population. In other words, we toss two dice. Figure 9.1 depicts this process in which we compute the mean for each sample. Because the value of the sample mean varies randomly
from sample to sample, we can regard X as a new random variable created by sampling.
Table 9.1 lists all the possible samples and their corresponding values of x.
FIGURE
9.1
Drawing Samples of Size 2 from a Population
Population
1, 1 x– = 1.0
Infinitely many
1's, 2's, . . . , 6's
1, 2 x– = 1.5
.
.
.
Parameters: m = 3.5
s 2 = 2.92
TABLE
9.1
6, 6 x– = 6.0
All Samples of Size 2 and Their Means
SAMPLE
x
SAMPLE
x
SAMPLE
x
1, 1
1.0
3, 1
2.0
5, 1
3.0
1, 2
1.5
3, 2
2.5
5, 2
3.5
1, 3
2.0
3, 3
3.0
5, 3
4.0
1, 4
2.5
3, 4
3.5
5, 4
4.5
1, 5
3.0
3, 5
4.0
5, 5
5.0
1, 6
3.5
3, 6
4.5
5, 6
5.5
2, 1
1.5
4, 1
2.5
6, 1
3.5
2, 2
2.0
4, 2
3.0
6, 2
4.0
2, 3
2.5
4, 3
3.5
6, 3
4.5
2, 4
3.0
4, 4
4.0
6, 4
5.0
2, 5
3.5
4, 5
4.5
6, 5
5.5
2, 6
4.0
4, 6
5.0
6, 6
6.0
There are 36 different possible samples of size 2; because each sample is equally likely,
the probability of any one sample being selected is 1/36. However, x can assume only 11
different possible values: 1.0, 1.5, 2.0, . . . , 6.0, with certain values of x occurring more frequently than others. The value x = 1.0 occurs only once, so its probability is 1/36.
The value x = 1.5 can occur in two ways—(1, 2) and (2, 1)—each having the same
probability (1/36). Thus, P(x = 1.5) = 2>36. The probabilities of the other values of x
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CHAPTER 9
are determined in similar fashion, and the resulting sampling distribution of the
sample mean is shown in Table 9.2.
TABLE
9. 2
—
Sampling Distribution of X
—
x
P (—
x)
1.0
1/36
1.5
2/36
2.0
3/36
2.5
4/36
3.0
5/36
3.5
6/36
4.0
5/36
4.5
4/36
5.0
3/36
5.5
2/36
6.0
1/36
The most interesting aspect of the sampling distribution of X is how different it is
from the distribution of X, as can be seen in Figure 9.2.
9.2
FIGURE
—
Distributions of X and X
p(x)
p(x–)
1
–
6
1
2
3
(a) Distribution of X
4
5
6
x
6
––
36
4
––
36
2
––
36
1
2
3
4
5
6
x–
(b) Sampling distribution of X
We can also compute the mean, variance, and standard deviation of the sampling
distribution. Once again using the definitions of expected value and variance, we determine the following parameters of the sampling distribution.
Mean of the sampling distribution of X:
mx = a xP(x)
= 1.0(1>36) + 1.5(2>36) + Á + 6.0(1>36)
= 3.5
Notice that the mean of the sampling distribution of X is equal to the mean of the population of the toss of a die computed previously.
Variance of the sampling distribution of X:
s2x = a (x - mx)2P(x)
= (1.0 - 3.5)2(1>36) + (1.5 - 3.5)2(2>36) + Á + (6.0 - 3.5)2(1>36)
= 1.46
It is no coincidence that the variance of the sampling distribution of X is exactly half
of the variance of the population of the toss of a die (computed previously as 2 2.92).
Standard deviation of the sampling distribution of X:
sx = 2s2x = 21.46 = 1.21
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311
SAMPLING DISTRIBUTIONS
It is important to recognize that the distribution of X is different from the distribution of X as depicted in Figure 9.2. However, the two random variables are related.
Their means are the same (mx = m = 3.5) and their variances are related (s2x = s2/2).
Don’t get lost in the terminology and notation. Remember that m and 2 are the
parameters of the population of X. To create the sampling distribution of X, we repeatedly drew samples of size n 2 from the population and calculated x for each sample.
Thus, we treat X as a brand-new random variable, with its own distribution, mean, and
variance. The mean is denoted mx, and the variance is denoted s2x.
If we now repeat the sampling process with the same population but with other values of n, we produce somewhat different sampling distributions of X. Figure 9.3 shows
the sampling distributions of X when n 5, 10, and 25.
FIGURE
9.3
—
Sampling Distributions of X for n ⴝ 5, 10, and 25
.6
.5
p(x–)
.4
.3
.2
.1
x
0
1
3.5
6
1
3.5
6
1
3.5
6
.8
.7
.6
p(x–)
.5
.4
.3
.2
.1
x
0
1.4
1.2
1
p(x–)
.8
.6
.4
.2
x
0
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For each value of n, the mean of the sampling distribution of X is the mean of the
population from which we’re sampling; that is,
mx = m = 3.5
The variance of the sampling distribution of the sample mean is the variance of the
population divided by the sample size:
s2x =
s2
n
The standard deviation of the sampling distribution is called the standard error of
the mean; that is,
sx =
s
2n
As you can see, the variance of the sampling distribution of X is less than the variance of the population we’re sampling from all sample sizes. Thus, a randomly selected
value of X (the mean of the number of spots observed in, say, five throws of the die) is
likely to be closer to the mean value of 3.5 than is a randomly selected value of X (the
number of spots observed in one throw). Indeed, this is what you would expect, because
in five throws of the die you are likely to get some 5s and 6s and some 1s and 2s, which
will tend to offset one another in the averaging process and produce a sample mean reasonably close to 3.5. As the number of throws of the die increases, the probability that
the sample mean will be close to 3.5 also increases. Thus, we observe in Figure 9.3 that
the sampling distribution of X becomes narrower (or more concentrated about the
mean) as n increases.
Another thing that happens as n gets larger is that the sampling distribution of x
becomes increasingly bell shaped. This phenomenon is summarized in the central
limit theorem.
Central Limit Theorem
The sampling distribution of the mean of a random sample drawn from any
population is approximately normal for a sufficiently large sample size. The
larger the sample size, the more closely the sampling distribution of X will
resemble a normal distribution.
The accuracy of the approximation alluded to in the central limit theorem depends
on the probability distribution of the population and on the sample size. If the population is normal, then X is normally distributed for all values of n. If the population is
nonnormal, then X is approximately normal only for larger values of n. In many practical situations, a sample size of 30 may be sufficiently large to allow us to use the normal
distribution as an approximation for the sampling distribution of X. However, if the
population is extremely nonnormal (for example, bimodal and highly skewed distributions), the sampling distribution will also be nonnormal even for moderately large
values of n.
Sampling Distribution of the Mean of Any Population We can extend the discoveries we’ve made to all infinitely large populations. Statisticians have shown that the
mean of the sampling distribution is always equal to the mean of the population and that
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SAMPLING DISTRIBUTIONS
313
the standard error is equal to s> 2n for infinitely large populations. (In Keller’s website
Appendix Using the Laws of Expected Value and Variance to Derive the Parameters of
Sampling Distributions we describe how to mathematically prove that mx = m and
s2x = s2>n.) However, if the population is finite the standard error is
sx =
N - n
A
N
- 1
2n
s
- n
where N is the population size and 3N
N - 1 is called the finite population correction
factor. (The source of the correction factor is provided in Keller’s website Appendix
Hypergeometric Distribution.) An analysis (see Exercises 9.13 and 9.14) reveals that if
the population size is large relative to the sample size, then the finite population correction factor is close to 1 and can be ignored. As a rule of thumb, we will treat any population that is at least 20 times larger than the sample size as large. In practice, most
applications involve populations that qualify as large because if the population is small,
it may be possible to investigate each member of the population, and in so doing, calculate the parameters precisely. As a consequence, the finite population correction factor
is usually omitted.
We can now summarize what we know about the sampling distribution of the
sample mean for large populations.
Sampling Distribution of the Sample Mean
1. mx = m
2. sx2 = s2>n and sx = s> 2n
3. If X is normal, then X is normal. If X is nonnormal, then X is
approximately normal for sufficiently large sample sizes. The definition
of “sufficiently large” depends on the extent of nonnormality of X.
Creating the Sampling Distribution Empirically
In the previous analysis, we created the sampling distribution of the mean theoretically. We did so by listing all the possible samples of size 2 and their probabilities.
(They were all equally likely with probability 1/36.) From this distribution, we produced the sampling distribution. We could also create the distribution empirically by
actually tossing two fair dice repeatedly, calculating the sample mean for each sample,
counting the number of times each value of X occurs, and computing the relative frequencies to estimate the theoretical probabilities. If we toss the two dice a large
enough number of times, the relative frequencies and theoretical probabilities (computed previously) will be similar. Try it yourself. Toss two dice 500 times, calculate
the mean of the two tosses, count the number of times each sample mean occurs, and
construct the histogram representing the sampling distribution. Obviously, this
approach is far from ideal because of the excessive amount of time required to toss the
dice enough times to make the relative frequencies good approximations for the theoretical probabilities. However, we can use the computer to quickly simulate tossing
dice many times.
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S E E I N G S TAT I S T I C S
applet 9 Fair Dice 1
This applet has two parts. The first part
simulates the tossing of one fair die.
You can toss 1 at a time, 10 at a time, or
100 at a time. The histogram of the
cumulative results is shown. The second
part allows you to simulate tossing
2 dice one set at a time, 10 sets a time,
or 100 sets a time. The histogram of the
means of the cumulative results is
exhibited. To start again, click Refresh or
Reload on the browser menu. The value
theoretical distribution displayed in
Figure 9.2? Discuss the reason for
N represents the number of sets. The
the deviations.
larger the value of N, the closer the
histogram approximates the theoretical
9.2 Does the simulated sampling
distribution of the mean of two dice
distribution.
Applet Exercises
Simulate 2,500 tosses of one fair die
distribution of one die look like the
one die and the sampling
distribution of the mean of two
dice appear to be about the same?
displayed in Figure 9.2? Discuss the
9.5 Do the spreads of the distribution
9.3 Do the distribution of one die and
9.1 Does the simulated probability
characterize the difference?
9.4 Do the centers of the distribution of
look like the theoretical distribution
reason for the deviations.
and 2,500 tosses of two fair dice.
different shapes? How would you
of one die and the sampling
distribution of the mean of two
the sampling distribution of the
dice appear to be about the same?
mean of two dice have the same or
Which one has the smaller spread?
S E E I N G S TAT I S T I C S
applet 10 Fair Dice 2
This applet allows you to simulate
Applet Exercises
tossing 12 fair dice and drawing the
Simulate 2,500 tosses of 12 fair dice.
sampling distribution of the mean. As
10.1 Does the simulated sampling
was the case with the previous applet,
distribution of X appear to be bell
you can toss 1 set, 10 sets, or 100 sets.
shaped?
To start again, click Refresh or Reload
on the browser menu.
10.2 Does it appear that the simulated
sampling distribution of the mean
of 12 fair dice is narrower than that
of 2 fair dice? Explain why this is so.
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315
S E E I N G S TAT I S T I C S
applet 11 Loaded Dice
This applet has two parts. The first part
simulates the tossing of a loaded die.
“Loaded” refers to the inequality of the
probabilities of the six outcomes. You
can toss 1 at a time, 10 at a time, or
100 at a time. The second part allows
you to simulate tossing 12 loaded dice 1
set at a time, 10 sets a time, or 100 sets
at a time.
Applet Exercises
Simulate 2,500 tosses of 12 loaded dice.
of 12 loaded dice is greater than
Simulate 2,500 tosses of one loaded die.
11.3 Does it appear that the mean of
that for 12 fair dice? Explain why
11.1 Estimate the probability of each
the simulated sampling
distribution of X is equal to 3.5?
value of X.
11.2 Use the estimated probabilities to
11.4 Does it appear that the standard
this is so.
11.5 Does the simulated sampling
distribution of the mean of 12
compute the expected value, vari-
deviation of the simulated
loaded dice appear to be bell
ance, and standard deviation of X.
sampling distribution of the mean
shaped? Explain why this is so.
S E E I N G S TAT I S T I C S
applet 12 Skewed Dice
This applet has two parts. The first part
simulates the tossing of a skewed die.
You can toss it 1 at a time, 10 at a time,
or 100 at a time. The second part allows
you to simulate tossing 2 dice 1 set at a
time, 10 sets a time, or 100 sets at a
time
Applet Exercises
Simulate 2,500 tosses of one skewed die.
12.1 Estimate the probability of each
value of X.
12.2 Use the estimated probabilities to
Simulate 2,500 tosses of 12 skewed dice.
12.3. Does it appear that the mean of
12.4. Does the simulated sampling
distribution of the mean of 12
compute the expected value, vari-
the simulated sampling
skewed dice appear to be bell
ance, and standard deviation of X.
distribution of X is less than 3.5?
shaped? Explain why this is so.
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CHAPTER 9
EXAMPLE
9. 1
Contents of a 32-Ounce Bottle
The foreman of a bottling plant has observed that the amount of soda in each 32-ounce
bottle is actually a normally distributed random variable, with a mean of 32.2 ounces
and a standard deviation of .3 ounce.
a. If a customer buys one bottle, what is the probability that the bottle will contain
more than 32 ounces?
b. If a customer buys a carton of four bottles, what is the probability that the mean
amount of the four bottles will be greater than 32 ounces?
SOLUTION
a. Because the random variable is the amount of soda in one bottle, we want to find
P(X 7 32), where X is normally distributed, 32.2, and .3. Hence,
P(X 7 32) = Pa
X - m
32 - 32.2
7
b
s
.3
= P(Z 7 - .67)
= 1 - P(Z 6 - .67)
= 1 - .2514 = .7486
b. Now we want to find the probability that the mean amount of four filled bottles
exceeds 32 ounces; that is, we want P(X 7 32). From our previous analysis and
from the central limit theorem, we know the following:
1. X is normally distributed.
2. mx = m = 32.2
3. sx = s> 2n = .3> 24 = .15
Hence,
P(X 7 32) = P ¢
X - mx
7
sx
32 - 32.2
≤ = P(Z 7 - 1.33)
.15
= 1 - P(Z 6 - 1.33) = 1 - .0918 = .9082
Figure 9.4 illustrates the distributions used in this example.
FIGURE
9.4
—
Distribution of X and Sampling Distribution of X
.7486
x
31.3
31.6
31.9 m = 32.2 32.5
32
32.8
33.1
.9082
x–
31.3
31.6
31.9 m–x = 32.2 32.5
32
32.8
33.1
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317
In Example 9.l(b), we began with the assumption that both and were known.
Then, using the sampling distribution, we made a probability statement about X.
Unfortunately, the values of and are not usually known, so an analysis such as that
in Example 9.1 cannot usually be conducted. However, we can use the sampling distribution to infer something about an unknown value of on the basis of a sample mean.
We want to find the probability that the sample mean is less than $750. Thus, we seek
P(X 6 750)
The distribution of X, the weekly income, is likely to be positively skewed, but not sufficiently so to
make the distribution of X nonnormal. As a result, we may assume that X is normal with mean
© Leland Bobbe/Getty Images
Salaries of a Business School’s Graduates: Solution
mx = m = 800 and standard deviation sx = s> 2n = 100> 225 = 20. Thus,
P(X 6 750) = P ¢
X - mx
sx
6
750 - 800
≤ = P(Z 6 - 2.5) = .0062
20
Figure 9.5 illustrates the distribution.
9.5
FIGURE
—
P ( X 750)
.0062
x–
750
800
The probability of observing a sample mean as low as $750 when the population mean is
$800 is extremely small. Because this event is quite unlikely, we would have to conclude that the
dean’s claim is not justified.
Using the Sampling Distribution for Inference
Our conclusion in the chapter-opening example illustrates how the sampling distribution can be used to make inferences about population parameters. The first form of
inference is estimation, which we introduce in the next chapter. In preparation for this
momentous occasion, we’ll present another way of expressing the probability associated
with the sampling distribution.
Recall the notation introduced in Section 8.2 (see page 278). We defined zA to be
the value of z such that the area to the right of zA under the standard normal curve is
equal to A. We also showed that z.025 1.96. Because the standard normal distribution
is symmetric about 0, the area to the left of 1.96 is also .025. The area between
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CHAPTER 9
1.96 and 1.96 is .95. Figure 9.6 depicts this notation. We can express the notation
algebraically as
P(- 1.96 6 Z 6 1.96) = .95
FIGURE
9.6
P (1.96 Z 1.96) .05
.95
.025
.025
z
0
–1.96
1.96
In this section, we established that
Z =
X - m
s> 2n
is standard normally distributed. Substituting this form of Z into the previous probability statement, we produce
X - m
P ¢ - 1.96 6
s> 2n
6 1.96 ≤ = .95
With a little algebraic manipulation (multiply all three terms by s> 2n and add to all three terms), we determine
P ¢ m - 1.96
s
2n
6 X 6 m + 1.96
s
2n
≤ = .95
Returning to the chapter-opening example where 800, 100, and n 25,
we compute
P ¢ 800 - 1.96
100
225
6 X 6 800 + 1.96
100
225
≤ = .95
Thus, we can say that
P(760.8 6 X 6 839.2) = .95
This tells us that there is a 95% probability that a sample mean will fall between 760.8
and 839.2. Because the sample mean was computed to be $750, we would have to conclude that the dean’s claim is not supported by the statistic.
Changing the probability from .95 to .90 changes the probability statement to
P ¢ m - 1.645
s
2n
6 X 6 m + 1.645
s
2n
≤ = .90
We can also produce a general form of this statement:
P ¢ m - za>2
s
2n
6 X 6 m + za>2
s
2n
≤ = 1 - a
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319
In this formula a (Greek letter alpha) is the probability that X does not fall into the
interval. To apply this formula, all we need do is substitute the values for , , n, and .
For example, with 800, 100, n 25, and .01, we produce
P ¢ m - z.005
s
2n
P ¢ 800 - 2.575
6 X 6 m + z.005
100
225
s
2n
≤ = 1 - .01
6 X 6 800 + 2.575
100
225
≤ = .99
P(748.5 6 X 6 851.5) = .99
which is another probability statement about X. In Section 10.2, we will use a similar
type of probability statement to derive the first statistical inference technique.
EXERCISES
9.1 Let X represent the result of the toss of a fair die.
Find the following probabilities.
a. P(X = 1)
b. P(X = 6)
9.2 Let X represent the mean of the toss of two fair dice.
Use the probabilities listed in Table 9.2 to determine
the following probabilities.
a. P(X = 1)
b. P(X = 6)
9.3 An experiment consists of tossing five balanced dice.
Find the following probabilities. (Determine the
exact probabilities as we did in Tables 9.1 and 9.2 for
two dice.)
a. P(X = 1)
b. P(X = 6)
9.4 Refer to Exercises 9.1 to 9.3. What do the probabil-
ities tell you about the variances of X and X?
9.5 A normally distributed population has a mean of 40
and a standard deviation of 12. What does the central limit theorem say about the sampling distribution of the mean if samples of size 100 are drawn
from this population?
9.6 Refer to Exercise 9.5. Suppose that the population is
not normally distributed. Does this change your
answer? Explain.
9.7 A sample of n 16 observations is drawn from a
normal population with 1,000 and 200.
Find the following.
a. P(X 7 1,050)
b. P(X 6 960)
c. P(X 7 1,100)
9.8 Repeat Exercise 9.7 with n 25.
9.9 Repeat Exercise 9.7 with n 100.
9.10 Given a normal population whose mean is 50 and
whose standard deviation is 5, find the probability
that a random sample of
a. 4 has a mean between 49 and 52.
b. 16 has a mean between 49 and 52.
c. 25 has a mean between 49 and 52.
9.11 Repeat Exercise 9.10 for a standard deviation of 10.
9.12 Repeat Exercise 9.10 for a standard deviation of 20.
9.13 a. Calculate the finite population correction factor
when the population size is N 1,000 and the
sample size is n 100.
b. Repeat part (a) when N 3,000.
c. Repeat part (a) when N 5,000.
d. What have you learned about the finite population
correction factor when N is large relative to n?
9.14 a. Suppose that the standard deviation of a popula-
tion with N 10,000 members is 500. Determine
the standard error of the sampling distribution of
the mean when the sample size is 1,000.
b. Repeat part (a) when n 500.
c. Repeat part (a) when n 100.
9.15 The heights of North American women are nor-
mally distributed with a mean of 64 inches and a
standard deviation of 2 inches.
a. What is the probability that a randomly selected
woman is taller than 66 inches?
b. A random sample of four women is selected.
What is the probability that the sample mean
height is greater than 66 inches?
c. What is the probability that the mean height of a
random sample of 100 women is greater than
66 inches?
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CHAPTER 9
9.16 Refer to Exercise 9.15. If the population of women’s
heights is not normally distributed, which, if any, of
the questions can you answer? Explain.
c. What is the probability that, in a random sample
of five North American adults, all watch television for more than 7 hours per day?
9.17 An automatic machine in a manufacturing process is
9.22 The manufacturer of cans of salmon that are supposed
operating properly if the lengths of an important
subcomponent are normally distributed with
mean 117 cm and standard deviation 5.2 cm.
a. Find the probability that one selected subcomponent is longer than 120 cm.
b. Find the probability that if four subcomponents
are randomly selected, their mean length exceeds
120 cm.
c. Find the probability that if four subcomponents
are randomly selected, all four have lengths that
exceed 120 cm.
to have a net weight of 6 ounces tells you that the net
weight is actually a normal random variable with a
mean of 6.05 ounces and a standard deviation of
.18 ounces. Suppose that you draw a random sample of
36 cans.
a. Find the probability that the mean weight of the
sample is less than 5.97 ounces.
b. Suppose your random sample of 36 cans of
salmon produced a mean weight that is less than
5.97 ounces. Comment on the statement made
by the manufacturer.
9.18 The amount of time the university professors devote
9.23 The number of customers who enter a supermarket
to their jobs per week is normally distributed with a
mean of 52 hours and a standard deviation of
6 hours.
a. What is the probability that a professor works for
more than 60 hours per week?
b. Find the probability that the mean amount of
work per week for three randomly selected professors is more than 60 hours.
c. Find the probability that if three professors are
randomly selected all three work for more than
60 hours per week.
each hour is normally distributed with a mean of 600
and a standard deviation of 200. The supermarket is
open 16 hours per day. What is the probability that
the total number of customers who enter the supermarket in one day is greater than 10,000? (Hint:
Calculate the average hourly number of customers
necessary to exceed 10,000 in one 16-hour day.)
9.19 The number of pizzas consumed per month by uni-
versity students is normally distributed with a mean
of 10 and a standard deviation of 3.
a. What proportion of students consume more than
12 pizzas per month?
b. What is the probability that in a random sample
of 25 students more than 275 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 25 students?)
9.20 The marks on a statistics midterm test are normally
distributed with a mean of 78 and a standard deviation of 6.
a. What proportion of the class has a midterm mark
of less than 75?
b. What is the probability that a class of 50 has an
average midterm mark that is less than 75?
9.21 The amount of time spent by North American
adults watching television per day is normally distributed with a mean of 6 hours and a standard deviation of 1.5 hours.
a. What is the probability that a randomly selected
North American adult watches television for
more than 7 hours per day?
b. What is the probability that the average time
watching television by a random sample of five
North American adults is more than 7 hours?
9.24 The sign on the elevator in the Peters Building,
which houses the School of Business and Economics
at Wilfrid Laurier University, states, “Maximum
Capacity 1,140 kilograms (2,500 pounds) or
16 Persons.” A professor of statistics wonders what
the probability is that 16 persons would weigh more
than 1,140 kilograms. Discuss what the professor
needs (besides the ability to perform the calculations) in order to satisfy his curiosity.
9.25 Refer to Exercise 9.24. Suppose that the professor dis-
covers that the weights of people who use the elevator
are normally distributed with an average of
75 kilograms and a standard deviation of 10 kilograms.
Calculate the probability that the professor seeks.
9.26 The time it takes for a statistics professor to mark
his midterm test is normally distributed with a mean
of 4.8 minutes and a standard deviation of 1.3 minutes. There are 60 students in the professor’s class.
What is the probability that he needs more than
5 hours to mark all the midterm tests? (The
60 midterm tests of the students in this year’s class
can be considered a random sample of the many
thousands of midterm tests the professor has marked
and will mark.)
9.27 Refer to Exercise 9.26. Does your answer change if
you discover that the times needed to mark a
midterm test are not normally distributed?
9.28 The restaurant in a large commercial building
provides coffee for the building’s occupants. The
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SAMPLING DISTRIBUTIONS
restaurateur has determined that the mean number of
cups of coffee consumed in a day by all the occupants
is 2.0 with a standard deviation of .6. A new tenant of
the building intends to have a total of 125 new
employees. What is the probability that the new
employees will consume more than 240 cups per day?
9. 2 S A M PL I N G D I S T R I B U T I O N
OF A
321
9.29 The number of pages produced by a fax machine in
a busy office is normally distributed with a mean of
275 and a standard deviation of 75. Determine the
probability that in 1 week (5 days) more than 1,500
faxes will be received?
P RO P ORT I O N
In Section 7.4, we introduced the binomial distribution whose parameter is p, the probability of success in any trial. In order to compute binomial probabilities, we assumed
that p was known. However, in the real world p is unknown, requiring the statistics
practitioner to estimate its value from a sample. The estimator of a population proportion of successes is the sample proportion; that is, we count the number of successes in
a sample and compute
X
PN =
n
(PN is read as p hat ) where X is the number of successes and n is the sample size. When
we take a sample of size n, we’re actually conducting a binomial experiment; as a result,
X is binomially distributed. Thus, the probability of any value of PN can be calculated
from its value of X. For example, suppose that we have a binomial experiment with
n 10 and p .4. To find the probability that the sample proportion PN is less than or
equal to .50, we find the probability that X is less than or equal to 5 (because
5/10 .50). From Table 1 in Appendix B we find with n 10 and p .4
P(PN … .50) = P(X … 5) = .8338
We can calculate the probability associated with other values of PN similarly.
Discrete distributions such as the binomial do not lend themselves easily to the
kinds of calculation needed for inference. And inference is the reason we need sampling
distributions. Fortunately, we can approximate the binomial distribution by a normal
distribution.
What follows is an explanation of how and why the normal distribution can be used
to approximate a binomial distribution. Disinterested readers can skip to page 325,
where we present the approximate sampling distribution of a sample proportion.
(Optional) Normal Approximation to the Binomial Distribution
Recall how we introduced continuous probability distributions in Chapter 8. We developed the density function by converting a histogram so that the total area in the rectangles equaled 1. We can do the same for a binomial distribution. To illustrate, let X be a
binomial random variable with n 20 and p .5. We can easily determine the probability of each value of X, where X 0, 1, 2, . . . , 19, 20. A rectangle representing a value
of x is drawn so that its area equals the probability. We accomplish this by letting the
height of the rectangle equal the probability and the base of the rectangle equal 1.
Thus, the base of each rectangle for x is the interval x .5 to x .5. Figure 9.7 depicts
this graph. As you can see, the rectangle representing x 10 is the rectangle whose
base is the interval 9.5 to 10.5 and whose height is P(X 10) .1762.
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FIGURE
9.7
Binomial Distribution with n 20 and p .5
P(x)
.20
.15
.10
.05
x
0
FIGURE
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
9.8
Binomial Distribution with n 20 and p .5 and Normal
Approximation
P(x)
.20
.15
.10
.05
0
x
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
9.5 10.5
If we now smooth the ends of the rectangles, we produce a bell-shaped curve as
seen in Figure 9.8. Thus, to use the normal approximation, all we need do is find the
area under the normal curve between 9.5 and 10.5.
To find normal probabilities requires us to first standardize x by subtracting the
mean and dividing by the standard deviation. The values for and are derived from
the binomial distribution being approximated. In Section 7.4 we pointed out that
m = np
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323
and
s = 2np(1 - p)
For n 20 and p .5, we have
m = np = 20(.5) = 10
and
s = 2np(1 - p) = 220(.5)(1 - .5) = 2.24
To calculate the probability that X 10 using the normal distribution requires that
we find the area under the normal curve between 9.5 and 10.5; that is,
P(X = 10) L P(9.5 6 Y 6 10.5)
where Y is a normal random variable approximating the binomial random variable X.
We standardize Y and use Table 3 of Appendix B to find
P(9.5 6 Y 6 10.5) = P ¢
Y - m
10.5 - 10
9.5 - 10
6
6
≤
2.24
s
2.24
= P(-.22 6 Z 6 .22) = (Z 6 .22) - P(Z 6 - .22)
= .5871 - .4129 = .1742
The actual probability that X equals 10 is
P(X = 10) = .1762
As you can see, the approximation is quite good.
Notice that to draw a binomial distribution, which is discrete, it was necessary to
draw rectangles whose bases were constructed by adding and subtracting .5 to the values of X. The .5 is called the continuity correction factor.
The approximation for any other value of X would proceed in the same manner. In
general, the binomial probability P(X x) is approximated by the area under a normal
curve between x .5 and x .5. To find the binomial probability P(X … x), we calculate
the area under the normal curve to the left of x .5. For the same binomial random variable, the probability that its value is less than or equal to 8 is P(X … 8) = .2517. The
normal approximation is
P(X … 8) L P(Y 6 8.5) = P ¢
Y - m
8.5 - 10
6
≤ = P(Z 6 - .67) = .2514
s
2.24
We find the area under the normal curve to the right of x .5 to determine the binomial
probability P(X Ú x). To illustrate, the probability that the binomial random variable
(with n 20 and p .5) is greater than or equal to 14 is P(X Ú 14) = .0577. The normal approximation is
P(X Ú 14) L P(Y 7 13.5) = P ¢
Y - m
13.5 - 10
7
≤ = P(Z 7 1.56) = .0594
s
2.24
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S E E I N G S TAT I S T I C S
applet 13 Normal Approximation to Binomial Probabilities
This applet shows how well the normal
distribution approximates the binomial
distribution. Select values for n and p,
which will specify a binomial distribution. Then set a value for k. The applet
calculates and graphs both the binomial
and normal probabilities for P(X … k).
Applet Exercises
13.1 Given a binomial distribution with
n 5 and p .2, use the applet to
compute the actual and normal
approximations of the following.
a. P(X … 0)
b. P(X … 1)
c. P(X … 2)
d. P(X … 3)
Describe how well the normal
distribution approximates the
binomial when n and p are small.
13.2 Repeat Exercise 13.1 with p .5.
Describe how well the normal
distribution approximates the
binomial when n is small and
when p is .5.
13.3 Suppose that X is a binomial
random variable with n 10 and
p .2. Use the applet to calculate
the actual and normal approximations of the following.
a. P(X … 2)
b. P(X … 3)
c. P(X … 4)
d. P(X … 5)
Describe how well the normal distribution approximates the binomial
when n 10 and when p is small.
13.4 Repeat Exercise 13.3 with p .5.
Describe how well the normal distribution approximates the binomial
when n 10 and when p is .5.
13.5 Describe the effect on the normal
approximation to the binomial as
n increases.
Omitting the Correction Factor for Continuity
When calculating the probability of individual values of X as we did when we computed
the probability that X equals 10 above, the correction factor must be used. If we don’t, we
are left with finding the area in a line, which is 0. When computing the probability of a
range of values of X, we can omit the correction factor. However, the omission of the correction factor will decrease the accuracy of the approximation. For example, if we
approximate P(X … 8) as we did previously except without the correction factor, we find
P(X … 8) L P(Y 6 8) = Pa
Y - m
8 - 10
6
b = P(Z 6 - .89) = .1867
s
2.24
The absolute size of the error between the actual cumulative binomial probability
and its normal approximation is quite small when the values of x are in the tail regions
of the distribution. For example, the probability that a binomial random variable with
n 20 and p .5 is less than or equal to 3 is
P(X … 3) = .0013
The normal approximation with the correction factor is
P(X … 3) L P(Y 6 3.5) = P ¢
Y - m
3.5 - 10
6
≤ = P(Z 6 - 2.90) = .0019
s
2.24
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325
The normal approximation without the correction factor is (using Excel)
P(X … 3) L P(Y 6 3) = P ¢
Y - m
3 - 10
6
≤ = P(Z 6 - 3.13) = .0009
s
2.24
For larger values of n, the differences between the normal approximation with and
without the correction factor are small even for values of X near the center of the distribution. For example, the probability that a binomial random variable with
n 1000 and p .3 is less than or equal to 260 is
P(X … 260) = .0029 (using Excel)
The normal approximation with the correction factor is
P(X … 260) L P(Y 6 260.5) = Pa
Y - m
260.5 - 300
6
b = P(Z 6 - 2.73) = .0032
s
14.49
The normal approximation without the correction factor is
P(X … 260) L P(Y 6 260) = Pa
Y - m
260 - 300
6
b = P(Z 6 - 2.76) = .0029
s
14.49
As we pointed out, the normal approximation of the binomial distribution is made necessary by the needs of statistical inference. As you will discover, statistical inference
generally involves the use of large values of n, and the part of the sampling distribution
that is of greatest interest lies in the tail regions. The correction factor was a temporary
tool that allowed us to convince you that a binomial distribution can be approximated
by a normal distribution. Now that we have done so, we will use the normal approximation of the binomial distribution to approximate the sampling distribution of a sample
proportion, and in such applications the correction factor will be omitted.
Approximate Sampling Distribution of a Sample Proportion
Using the laws of expected value and variance (see Keller’s website Appendix Using the
Laws of Expected Value and Variance to Derive the Parameters of Sampling
Distributions), we can determine the mean, variance, and standard deviation of PN . We
will summarize what we have learned.
Sampling Distribution of a Sample Proportion
1. PN is approximately normally distributed provided that np and n(1 p)
are greater than or equal to 5.
2. The expected value: E(PN ) = p
3. The variance: V(PN ) = s2pN =
p(1 - p) *
n
4. The standard deviation: spN = 1p(1 - p)兾n
(The standard deviation of PN is called the standard error of the proportion.)
*As was the case with the standard error of the mean (page 313), the standard error of a proportion is
1p(1 - p)>n when sampling from infinitely large populations. When the population is finite, the standard error of the proportion must include the finite population correction factor, which can be omitted
when the population is large relative to the sample size, a very common occurrence in practice.
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The sample size requirement is theoretical because, in practice, much larger sample sizes are needed for the normal approximation to be useful.
EXAMPLE
Political Survey
9. 2
In the last election, a state representative received 52% of the votes cast. One year after
the election, the representative organized a survey that asked a random sample of
300 people whether they would vote for him in the next election. If we assume that his
popularity has not changed, what is the probability that more than half of the sample
would vote for him?
SOLUTION
The number of respondents who would vote for the representative is a binomial random variable with n 300 and p .52. We want to determine the probability that the
sample proportion is greater than 50%. In other words, we want to find P(PN 7 .50).
We now know that the sample proportion PN is approximately normally distributed
with mean p .52 and standard deviation = 1p(1 - p)兾n = 1(.52)(.48)兾300 = .0288.
Thus, we calculate
PN - p
.50 - .52
P(PN 7 .50) = Pa
7
b
.0288
1p(1 - p)兾n
= P(Z 7 - .69) = 1 - P(Z 6 - .69) = 1 - .2451 = .7549
If we assume that the level of support remains at 52%, the probability that more
than half the sample of 300 people would vote for the representative is .7549.
EXERCISES
Use the normal approximation without the correction factor to
find the probabilities in the following exercises.
9.30 a. In a binomial experiment with n 300 and
p .5, find the probability that PN is greater
than 60%.
b. Repeat part (a) with p .55.
c. Repeat part (a) with p .6
9.31 a. The probability of success on any trial of a bino-
mial experiment is 25%. Find the probability
that the proportion of successes in a sample of
500 is less than 22%.
b. Repeat part (a) with n 800.
c. Repeat part (a) with n 1,000.
9.32 Determine the probability that in a sample of 100
the sample proportion is less than .75 if p .80.
9.33 A binomial experiment where p .4 is conducted.
Find the probability that in a sample of 60 the proportion of successes exceeds .35.
9.34 The proportion of eligible voters in the next elec-
tion who will vote for the incumbent is assumed to
be 55%. What is the probability that in a random
sample of 500 voters less than 49% say they will vote
for the incumbent?
9.35 The assembly line that produces an electronic com-
ponent of a missile system has historically resulted in
a 2% defective rate. A random sample of 800 components is drawn. What is the probability that the
defective rate is greater than 4%? Suppose that in the
random sample the defective rate is 4%. What does that
suggest about the defective rate on the assembly line?
9.36 a. The manufacturer of aspirin claims that the pro-
portion of headache sufferers who get relief with
just two aspirins is 53%. What is the probability
that in a random sample of 400 headache sufferers, less than 50% obtain relief? If 50% of the
sample actually obtained relief, what does this
suggest about the manufacturer’s claim?
b. Repeat part (a) using a sample of 1,000.
9.37 The manager of a restaurant in a commercial build-
ing has determined that the proportion of customers
who drink tea is 14%. What is the probability that in
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the next 100 customers at least 10% will be tea
drinkers?
9.41 A psychologist believes that 80% of male drivers
when lost continue to drive hoping to find the location they seek rather than ask directions. To examine
this belief, he took a random sample of 350 male drivers and asked each what they did when lost. If the
belief is true, determine the probability that less
than 75% said they continue driving.
9.38 A commercial for a manufacturer of household
appliances claims that 3% of all its products require
a service call in the first year. A consumer protection
association wants to check the claim by surveying
400 households that recently purchased one of the
company’s appliances. What is the probability that
more than 5% require a service call within the first
year? What would you say about the commercial’s
honesty if in a random sample of 400 households 5%
report at least one service call?
9.42 The Red Lobster restaurant chain regularly sur-
veys its customers. On the basis of these surveys,
the management of the chain claims that 75% of its
customers rate the food as excellent. A consumer
testing service wants to examine the claim by asking 460 customers to rate the food. What is the
probability that less than 70% rate the food as
excellent?
9.39 The Laurier Company’s brand has a market share of
30%. Suppose that 1,000 consumers of the product
are asked in a survey which brand they prefer. What
is the probability that more than 32% of the respondents say they prefer the Laurier brand?
9.43 An accounting professor claims that no more than
one-quarter of undergraduate business students
will major in accounting. What is the probability
that in a random sample of 1,200 undergraduate
business students, 336 or more will major in
accounting?
9.40 A university bookstore claims that 50% of its cus-
tomers are satisfied with the service and prices.
a. If this claim is true, what is the probability that in
a random sample of 600 customers less than 45%
are satisfied?
b. Suppose that in a random sample of 600
customers, 270 express satisfaction with the
bookstore. What does this tell you about the
bookstore’s claim?
9. 3 S A M PL I N G D I S T R I B U T I O N
MEANS
327
9.44 Refer to Exercise 9.43. A survey of a random
OF THE
sample of 1,200 undergraduate business students
indicates that 336 students plan to major in
accounting. What does this tell you about the professor’s claim?
DIFFERENCE
BET WEEN
T WO
Another sampling distribution that you will soon encounter is that of the difference
between two sample means. The sampling plan calls for independent random samples
drawn from each of two normal populations. The samples are said to be independent if
the selection of the members of one sample is independent of the selection of the members of the second sample. We will expand upon this discussion in Chapter 13. We are
interested in the sampling distribution of the difference between the two sample means.
In Section 9.1, we introduced the central limit theorem, which states that in
repeated sampling from a normal population whose mean is and whose standard deviation is , the sampling distribution of the sample mean is normal with mean and
standard deviation s> 1n. Statisticians have shown that the difference between two
independent normal random variables is also normally distributed. Thus, the difference
between two sample means X1 - X2 is normally distributed if both populations are
normal.
Through the use of the laws of expected value and variance we derive the expected
value and variance of the sampling distribution of X1 - X2:
mx
1 - x2
= m1 - m2
and
s2x - x =
1
2
s21
n1
+
s22
n2
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Thus, it follows that in repeated independent sampling from two populations with
means m1 and m2 and standard deviations 1 and 2, respectively, the sampling distribution of X1 - X2 is normal with mean
mx
1 - x2
= m1 - m2
and standard deviation (which is the standard error of the difference between two
means)
sx
1 - x2
=
s21
B n1
+
s22
n2
If the populations are nonnormal, then the sampling distribution is only approximately normal for large sample sizes. The required sample sizes depend on the extent of
nonnormality. However, for most populations, sample sizes of 30 or more are sufficient.
Figure 9.9 depicts the sampling distribution of the difference between two means.
FIGURE
9.9
—
—
Sampling Distribution of X 1 ⴚ X 2
x1 – x2
m1 – m2
EXAMPLE
9. 3
Starting Salaries of MBAs
Suppose that the starting salaries of MBAs at Wilfrid Laurier University (WLU) are
normally distributed, with a mean of $62,000 and a standard deviation of $14,500. The
starting salaries of MBAs at the University of Western Ontario (UWO) are normally
distributed, with a mean of $60,000 and a standard deviation of $18,300. If a random
sample of 50 WLU MBAs and a random sample of 60 UWO MBAs are selected, what
is the probability that the sample mean starting salary of WLU graduates will exceed
that of the UWO graduates?
SOLUTION
We want to determine P(X1 - X2 7 0). We know that X1 - X2 is normally distributed
with mean m1 - m2 = 62,000 - 60,000 = 2,000 and standard deviation
s21
B n1
+
s22
n2
=
14,5002
18,3002
+
= 3,128
B 50
60
We can standardize the variable and refer to Table 3 of Appendix B:
P(X 1 - X 2 7 0) = P ±
(X 1 - X 2) - (m1 - m2)
s21
B n1
+
s22
7
0 - 2,000
≤
3,128
n2
= P(Z 7 - .64) = 1 - P(Z 6 - .64) = 1 - .2611 = .7389
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329
There is a .7389 probability that for a sample of size 50 from the WLU graduates
and a sample of size 60 from the UWO graduates, the sample mean starting salary of
WLU graduates will exceed the sample mean of UWO graduates.
S E E I N G S TAT I S T I C S
applet 14 Distribution of the Differences between Means
The first part of this applet depicts two
graphs. The first graph shows the
distribution of the random variable of
two populations. Moving the top slider
shifts the first distribution left or right.
The right slider controls the value of the
population standard deviations, which
are assumed to be equal. By moving
each slider, you can see the relationship
between the two populations.
The second graph describes the
sampling distribution of the mean of
each population in the first graph.
relationship among the two population
Moving the right slider increases or
means, the common population
decreases the sample size, which is the
standard deviation, and the sample size.
same for both samples.
The second part of the applet has
three graphs. The first two graphs are
s1 = s2 = 1.1 to
s1 = s2 = 3.0 on the population
random variables, the sampling
Applet Exercises
distribution of x1, the sampling
14.1 Describe the effect of changing
distribution of x2, and the
the difference between population
sampling distribution of x1 - x2.
The third graph depicts the sampling
means from 5.0 to 4.5 on the
Describe what happened.
distribution of the difference between
population random variables, the
the two sample means from the
sampling distribution of x1, the
populations described previously.
sampling distribution of x2, and
n1 = n2 = 20 on the sampling
the sampling distribution of
distribution of x1, the sampling
x1 - x2. Describe what happened.
distribution of x2, and the sampling
identical to the graphs in the first part.
Moving the sliders allows you to
see the effect on the sampling
distribution of x1 - x2 of changing the
14.2 Describe the effect of changing
the standard deviations from
14.3 Describe the effect of changing the
sample sizes from n1 = n2 = 2 to
distribution of x1 - x2. Describe
the effect.
EXERCISES
9.45 Independent random samples of 10 observations
each are drawn from normal populations. The parameters of these populations are
Population 1: 280, 25
Population 2: 270, 30
Find the probability that the mean of sample 1 is
greater than the mean of sample 2 by more than 25.
9.46 Repeat Exercise 9.45 with samples of size 50.
9.47 Repeat Exercise 9.45 with samples of size 100.
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9.48 Suppose that we have two normal populations with
the means and standard deviations listed here. If
random samples of size 25 are drawn from each population, what is the probability that the mean of
sample 1 is greater than the mean of sample 2?
Population 1: 40, 6
Population 2: 38, 8
9.49 Repeat Exercise 9.48 assuming that the standard
deviations are 12 and 16, respectively.
9.50 Repeat Exercise 9.48 assuming that the means are
140 and 138, respectively.
9.51 A factory’s worker productivity is normally distrib-
uted. One worker produces an average of 75 units
per day with a standard deviation of 20. Another
worker produces at an average rate of 65 per day
with a standard deviation of 21. What is the probability that during one week (5 working days), worker
1 will outproduce worker 2?
9.52 A professor of statistics noticed that the marks in his
course are normally distributed. He has also noticed
that his morning classes average 73%, with a standard deviation of 12% on their final exams. His
afternoon classes average 77%, with a standard deviation of 10%. What is the probability that the mean
mark of four randomly selected students from a
morning class is greater than the average mark of
9. 4 F RO M H E R E
TO
four randomly selected students from an afternoon
class?
9.53 The manager of a restaurant believes that waiters
and waitresses who introduce themselves by telling
customers their names will get larger tips than those
who don’t. In fact, she claims that the average tip
for the former group is 18%, whereas that of the
latter is only 15%. If tips are normally distributed
with a standard deviation of 3%, what is the probability that in a random sample of 10 tips recorded
from waiters and waitresses who introduce themselves and 10 tips from waiters and waitresses who
don’t, the mean of the former will exceed that of the
latter?
9.54 The average North American loses an average of
15 days per year to colds and flu. The natural remedy echinacea reputedly boosts the immune system.
One manufacturer of echinacea pills claims that consumers of its product will reduce the number of days
lost to colds and flu by one-third. To test the claim, a
random sample of 50 people was drawn. Half took
echinacea, and the other half took placebos. If we
assume that the standard deviation of the number of
days lost to colds and flu with and without echinacea
is 3 days, find the probability that the mean number
of days lost for echinacea users is less than that for
nonusers.
INFERENCE
The primary function of the sampling distribution is statistical inference. To see how
the sampling distribution contributes to the development of inferential methods, we
need to briefly review how we got to this point.
In Chapters 7 and 8, we introduced probability distributions, which allowed us to
make probability statements about values of the random variable. A prerequisite of this
calculation is knowledge of the distribution and the relevant parameters. In Example 7.9,
we needed to know that the probability that Pat Statsdud guesses the correct answer is
20% ( p .2) and that the number of correct answers (successes) in 10 questions (trials) is a binomial random variable. We could then compute the probability of any number of successes. In Example 8.3, we needed to know that the return on investment is
normally distributed with a mean of 10% and a standard deviation of 5%. These three
bits of information allowed us to calculate the probability of various values of the
random variable.
Figure 9.10 symbolically represents the use of probability distributions. Simply
put, knowledge of the population and its parameter(s) allows us to use the probability distribution to make probability statements about individual members of
the population. The direction of the arrows indicates the direction of the flow
of information.
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FIGURE
9.10
Population
331
Probability Distribution
Probability distribution
Individual
& Parameter(s)
In this chapter, we developed the sampling distribution, wherein knowledge of the
parameter(s) and some information about the distribution allow us to make probability
statements about a sample statistic. In Example 9.1(b), knowing the population mean
and standard deviation and assuming that the population is not extremely nonnormal
enabled us to calculate a probability statement about a sample mean. Figure 9.11
describes the application of sampling distributions.
FIGURE
9.11
Population
Sampling Distribution
Sampling distribution
& Parameter(s)
Statistic
Notice that in applying both probability distributions and sampling distributions,
we must know the value of the relevant parameters, a highly unlikely circumstance. In
the real world, parameters are almost always unknown because they represent descriptive
measurements about extremely large populations. Statistical inference addresses this
problem. It does so by reversing the direction of the flow of knowledge in Figure 9.11.
In Figure 9.12, we display the character of statistical inference. Starting in
Chapter 10, we will assume that most population parameters are unknown. The statistics practitioner will sample from the population and compute the required statistic. The sampling distribution of that statistic will enable us to draw inferences about
the parameter.
FIGURE
9.12
Statistic
Sampling Distribution in Inference
Sampling distribution
Parameter
You may be surprised to learn that, by and large, that is all we do in the remainder of
this book. Why then do we need another 14 chapters? They are necessary because there
are many more parameter and sampling distribution combinations that define the inferential procedures to be presented in an introductory statistics course. However, they all
work in the same way. If you understand how one procedure is developed, then you will
likely understand all of them. Our task in the next two chapters is to ensure that you
understand the first inferential method. Your job is identical.
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CHAPTER 9
CHAPTER SUM M ARY
The sampling distribution of a statistic is created by
repeated sampling from one population. In this chapter, we
introduced the sampling distribution of the mean, the
proportion, and the difference between two means. We
described how these distributions are created theoretically
and empirically.
IMPORTANT TERMS
Sampling distribution 308
Sampling distribution of the sample mean 310
Standard error of the mean 312
Central limit theorem 312
Finite population correction factor 313
Sampling distribution of a sample proportion 321
Continuity correction factor 323
Standard error of the proportion 325
Difference between two sample means 327
Sampling distribution of X1 - X2 327
Standard error of the difference between two
means 328
SYMBOLS
Symbol
Pronounced
mx
s2x
mu x bar
sigma squared x bar
Mean of the sampling distribution of the sample mean
Variance of the sampling distribution of the sample
mean
sigma x bar
Standard deviation (standard error) of the sampling
distribution of the sample mean
alpha
Probability
p hat
Sample proportion
sigma squared p hat
Variance of the sampling distribution of the sample
proportion
sigma p hat
Standard deviation (standard error) of the sampling
distribution of the sample proportion
mu x bar 1 minus x bar 2
Mean of the sampling distribution of the difference
between two sample means
sigma squared x bar 1 minus x bar 2 Variance of the sampling distribution of the difference
between two sample means
sigma x bar 1 minus x bar 2
Standard deviation (standard error) of the sampling
distribution of the difference between two sample means
sx
PN
s2pN
spN
mx
1 - x2
s2x - x
1
sx
Represents
2
1 - x2
FORMULAS
Expected value of the sample mean
E(X ) = mx = m
Variance of the sample mean
s2
V(X ) = s2x =
n
Standard error of the sample mean
sx =
s
1n
Standardizing the sample mean
Z =
X - m
s兾 1n
Expected value of the sample proportion
E(PN ) = mpN = p
Variance of the sample proportion
p(1 - p)
V(PN ) = s2pN =
n
Standard error of the sample proportion
spN =
p(1 - p)
n
B
Standardizing the sample proportion
PN - p
Z =
1p(1 - p)兾n
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SAMPLING DISTRIBUTIONS
Expected value of the difference between two means
E(X1 - X2) = mx
1 - x2
= m1 - m2
Variance of the difference between two means
V(X1 - X2) = s2x - x =
1
2
s21
n1
+
s22
333
Standardizing the difference between two sample means
Z =
(X1 - X2) - (m1 - m2)
s21
B n1
+
s22
n2
n2
Standard error of the difference between two means
sx
x
1- 2
=
s21
B n1
+
s22
n2
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10
© age fotostock/Super Stock
INTRODUCTION TO
ESTIMATION
Concepts of Estimation
10.2
Estimating the Population Mean When the Population
Standard Deviation Is Known
10.3
Selecting the Sample Size
Determining the Sample Size to Estimate
the Mean Tree Diameter
A lumber company has just acquired the rights to a large tract of land containing thousands of trees.
A lumber company needs to be able to estimate the amount of lumber it can harvest in a
tract of land to determine whether the effort will be profitable. To do so, it must estimate the
mean diameter of the trees. It decides to estimate that parameter to within 1 inch with 90% confidence. A forester familiar with the territory guesses that the diameters of the trees are normally
distributed with a standard deviation of 6 inches. Using the formula on page 355, he determines
© Randy Mayor/Botanical/
Jupiterimages
10.1
See page 355 for the
solution.
that he should sample 98 trees. After sampling those 98 trees, the forester calculates the sample
mean to be 25 inches. Suppose that after he has completed his sampling and calculations, he
discovers that the actual standard deviation is 12 inches. Will he be satisfied with the result?
335
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CHAPTER 10
I N T RO D U C T I O N
10.1
CONC E P T S
H
aving discussed descriptive statistics (Chapter 4), probability distributions
(Chapters 7 and 8), and sampling distributions (Chapter 9), we are ready to
tackle statistical inference. As we explained in Chapter 1, statistical inference is the
process by which we acquire information and draw conclusions about populations from
samples. There are two general procedures for making inferences about populations:
estimation and hypothesis testing. In this chapter, we introduce the concepts and foundations of estimation and demonstrate them with simple examples. In Chapter 11, we
describe the fundamentals of hypothesis testing. Because most of what we do in the
remainder of this book applies the concepts of estimation and hypothesis testing, understanding Chapters 10 and 11 is vital to your development as a statistics practitioner.
OF
E S T I M AT I O N
As its name suggests, the objective of estimation is to determine the approximate value
of a population parameter on the basis of a sample statistic. For example, the sample
mean is employed to estimate the population mean. We refer to the sample mean as the
estimator of the population mean. Once the sample mean has been computed, its value
is called the estimate. In this chapter, we will introduce the statistical process whereby
we estimate a population mean using sample data. In the rest of the book, we use the
concepts and techniques introduced here for other parameters.
Point and Interval Estimators
We can use sample data to estimate a population parameter in two ways. First, we can
compute the value of the estimator and consider that value as the estimate of the parameter. Such an estimator is called a point estimator.
Point Estimator
A point estimator draws inferences about a population by estimating the
value of an unknown parameter using a single value or point.
There are three drawbacks to using point estimators. First, it is virtually certain
that the estimate will be wrong. (The probability that a continuous random variable will
equal a specific value is 0; that is, the probability that x will exactly equal ␮ is 0.) Second,
we often need to know how close the estimator is to the parameter. Third, in drawing
inferences about a population, it is intuitively reasonable to expect that a large sample
will produce more accurate results because it contains more information than a smaller
sample does. But point estimators don’t have the capacity to reflect the effects of larger
sample sizes. As a consequence, we use the second method of estimating a population
parameter, the interval estimator.
Interval Estimator
An interval estimator draws inferences about a population by estimating
the value of an unknown parameter using an interval.
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As you will see, the interval estimator is affected by the sample size; because it possesses this feature, we will deal mostly with interval estimators in this text.
To illustrate the difference between point and interval estimators, suppose that a
statistics professor wants to estimate the mean summer income of his second-year business students. Selecting 25 students at random, he calculates the sample mean weekly
income to be $400. The point estimate is the sample mean. In other words, he estimates
the mean weekly summer income of all second-year business students to be $400. Using
the technique described subsequently, he may instead use an interval estimate; he estimates that the mean weekly summer income of second-year business students to lie
between $380 and $420.
Numerous applications of estimation occur in the real world. For example, television network executives want to know the proportion of television viewers who are
tuned in to their networks; an economist wants to know the mean income of university graduates; and a medical researcher wishes to estimate the recovery rate of heart
attack victims treated with a new drug. In each of these cases, to accomplish the
objective exactly, the statistics practitioner would have to examine each member of
the population and then calculate the parameter of interest. For instance, network
executives would have to ask each person in the country what he or she is watching to
determine the proportion of people who are watching their shows. Because there are
millions of television viewers, the task is both impractical and prohibitively expensive.
An alternative would be to take a random sample from this population, calculate the
sample proportion, and use that as an estimator of the population proportion. The
use of the sample proportion to estimate the population proportion seems logical.
The selection of the sample statistic to be used as an estimator, however, depends on
the characteristics of that statistic. Naturally, we want to use the statistic with the
most desirable qualities for our purposes.
One desirable quality of an estimator is unbiasedness.
Unbiased Estimator
An unbiased estimator of a population parameter is an estimator whose
expected value is equal to that parameter.
This means that if you were to take an infinite number of samples and calculate the
value of the estimator in each sample, the average value of the estimators would equal
the parameter. This amounts to saying that, on average, the sample statistic is equal to
the parameter.
We know that the sample mean X is an unbiased estimator of the population mean ␮.
In presenting the sampling distribution of X in Section 9.1, we stated that E1X2 = m.
We also know that the sample proportion is an unbiased estimator of the population
proportion because E(PN ) = p and that the difference between two sample means is an
unbiased estimator of the difference between two population means because
E1X1 - X22 = m1 - m2.
Recall that in Chapter 4 we defined the sample variance as
1xi - x22
s2 = a
n - 1
At the time, it seemed odd that we divided by n ⫺ 1 rather than by n. The reason for
choosing n ⫺ 1 was to make E1s22 = s2 so that this definition makes the sample variance
an unbiased estimator of the population variance. (The proof of this statement requires
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about a page of algebraic manipulation, which is more than we would be comfortable
presenting here.) Had we defined the sample variance using n in the denominator, the
resulting statistic would be a biased estimator of the population variance, one whose
expected value is less than the parameter.
Knowing that an estimator is unbiased only assures us that its expected value equals
the parameter; it does not tell us how close the estimator is to the parameter. Another
desirable quality is that as the sample size grows larger, the sample statistic should come
closer to the population parameter. This quality is called consistency.
Consistency
An unbiased estimator is said to be consistent if the difference between the
estimator and the parameter grows smaller as the sample size grows larger.
The measure we use to gauge closeness is the variance (or the standard deviation).
Thus, X is a consistent estimator of ␮ because the variance of X is ␴2/n. This implies
that as n grows larger, the variance of X grows smaller. As a consequence, an increasing
proportion of sample means falls close to ␮.
Figure 10.1 depicts two sampling distributions of X when samples are drawn from
a population whose mean is 0 and whose standard deviation is 10. One sampling distribution is based on samples of size 25, and the other is based on samples of size 100. The
former is more spread out than the latter.
FIGURE
10.1
Sampling Distribution of X with n ⫽ 25 and n ⫽ 100
Sampling distribution
of X: n = 100
Sampling distribution
of X: n = 25
x–
m
Similarly, PN is a consistent estimator of p because it is unbiased and the variance of
NP is p11 - p2>n, which grows smaller as n grows larger.
A third desirable quality is relative efficiency, which compares two unbiased estimators of a parameter.
Relative Efficiency
If there are two unbiased estimators of a parameter, the one whose variance
is smaller is said to have relative efficiency.
We have already seen that the sample mean is an unbiased estimator of the population mean and that its variance is ␴2/n. In the next section, we will discuss the use of the
sample median as an estimator of the population mean. Statisticians have established that
the sample median is an unbiased estimator but that its variance is greater than that of
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the sample mean (when the population is normal). As a consequence, the sample mean is
relatively more efficient than the sample median when estimating the population mean.
In the remaining chapters of this book, we will present the statistical inference of a
number of different population parameters. In each case, we will select a sample statistic that is unbiased and consistent. When there is more than one such statistic, we will
choose the one that is relatively efficient to serve as the estimator.
Developing an Understanding of Statistical Concepts
In this section, we described three desirable characteristics of estimators: unbiasedness,
consistency, and relative efficiency. An understanding of statistics requires that you
know that there are several potential estimators for each parameter, but that we choose
the estimators used in this book because they possess these characteristics.
EXERCISES
10.1 How do point estimators and interval estimators differ?
sampling distribution of a consistent estimator when
the sample size increases.
10.2 Define unbiasedness.
10.3 Draw a sampling distribution of an unbiased esti-
mator.
10.4 Draw a sampling distribution of a biased estimator.
10.5 Define consistency.
10. 2
10.6 Draw diagrams representing what happens to the
10.7 Define relative efficiency.
10.8 Draw a diagram that shows the sampling distribu-
tion representing two unbiased estimators, one of
which is relatively efficient.
E S T I M AT I N G T H E P O P U L AT I O N M E A N W H E N
S TA N DA R D D E V I AT I O N I S K N O W N
THE
P O P U L AT I O N
We now describe how an interval estimator is produced from a sampling distribution.
We choose to demonstrate estimation with an example that is unrealistic. However, this
liability is offset by the example’s simplicity. When you understand more about estimation, you will be able to apply the technique to more realistic situations.
Suppose we have a population with mean ␮ and standard deviation ␴. The population
mean is assumed to be unknown, and our task is to estimate its value. As we just discussed,
the estimation procedure requires the statistics practitioner to draw a random sample of
size n and calculate the sample mean x.
The central limit theorem presented in Section 9.1 stated that X is normally distributed if X is normally distributed, or approximately normally distributed if X is nonnormal and n is sufficiently large. This means that the variable
Z =
X - m
s> 2n
is standard normally distributed (or approximately so). In Section 9.1 (page 318) we
developed the following probability statement associated with the sampling distribution
of the mean:
P a m - Za>2
s
2n
6 X 6 m + Za>2
s
2n
b = 1 - a
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which was derived from
X - m
Pa - Za>2 6
s> 2n
6 Za>2 b = 1 - a
Using a similar algebraic manipulation, we can express the probability in a slightly
different form:
P aX - Za>2
s
2n
6 m 6 X + Za>2
s
2n
b = 1 - a
Notice that in this form the population mean is in the center of the interval created
by adding and subtracting Z␣/2 standard errors to and from the sample mean. It is
important for you to understand that this is merely another form of probability statement about the sample mean. This equation says that, with repeated sampling from this
population, the proportion of values of X for which the interval
X - Za>2
s
2n
, X + Za>2
s
2n
includes the population mean ␮ is equal to 1 ⫺ ␣. This form of probability statement is
very useful to us because it is the confidence interval estimator of ␮.
Confidence Interval Estimator of ␮*
s
s
x - za>2
, x + za>2
2n
2n
The probability 1 ⫺ ␣ is called the confidence level.
s
x - za>2
is called the lower confidence limit (LCL).
2n
s
x + za>2
is called the upper confidence limit (UCL).
2n
We often represent the confidence interval estimator as
x ; za>2
s
2n
where the minus sign defines the lower confidence limit and the plus sign
defines the upper confidence limit.
To apply this formula, we specify the confidence level 1 ⫺ ␣, from which we determine ␣, ␣/2, z␣/2 (from Table 3 in Appendix B). Because the confidence level is the
*Since Chapter 7, we’ve been using the convention whereby an uppercase letter (usually X) represents a
random variable and a lowercase letter (usually x) represents one of its values. However, in the formulas
used in statistical inference, the distinction between the variable and its value becomes blurred.
Accordingly, we will discontinue the notational convention and simply use lowercase letters except when
we wish to make a probability statement.
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341
probability that the interval includes the actual value of ␮, we generally set 1 ⫺ ␣ close
to 1 (usually between .90 and .99).
In Table 10.1, we list four commonly used confidence levels and their associated
values of z␣/2. For example, if the confidence level is 1 ⫺ ␣ ⫽ .95, ␣ ⫽ .05, ␣/2 ⫽ .025,
and z␣/2 ⫽ z0.25 ⫽ 1.96. The resulting confidence interval estimator is then called the
95% confidence interval estimator of ␮.
TABLE
10.1
1⫺␣
Four Commonly Used Confidence Levels and z␣/2
␣
␣/2
Z␣/2
.90
.10
.05
z0.05 ⫽ 1.645
.95
.05
.025
z0.25 ⫽ 1.96
.98
.02
.01
z.01 ⫽ 2.33
.99
.01
.005
z.005 ⫽ 2.575
The following example illustrates how statistical techniques are applied. It also
illustrates how we intend to solve problems in the rest of this book. The solution
process that we advocate and use throughout this book is by and large the same one
that statistics practitioners use to apply their skills in the real world. The process is
divided into three stages. Simply stated, the stages are (1) the activities we perform
before the calculations, (2) the calculations, and (3) the activities we perform after the
calculations.
In stage 1, we determine the appropriate statistical technique to employ. Of course,
for this example you will have no difficulty identifying the technique because you know
only one at this point. (In practice, stage 1 also addresses the problem of how to gather
the data. The methods used in the examples, exercises, and cases are described in the
problem.)
In the second stage we calculate the statistics. We will do this in three ways.* To
illustrate how the computations are completed, we will do the arithmetic manually with
the assistance of a calculator. Solving problems by hand often provides insights into the
statistical inference technique. Additionally, we will use the computer in two ways.
First, in Excel we will use the Analysis ToolPak (Data menu item Data Analysis) or the
add-ins we created for this book (Add-Ins menu item Data Analysis Plus).
(Additionally, we will teach how to create do-it-yourself Excel spreadsheets that use
built-in statistical functions.) Finally, we will use Minitab, one of the easiest software
packages to use.
In the third and last stage of the solution, we intend to interpret the results and deal
with the question presented in the problem. To be capable of properly interpreting statistical results, one needs to have an understanding of the fundamental principles
underlying statistical inference.
*We anticipate that students in most statistics classes will use only one of the three methods of computing statistics: the choice made by the instructor. If such is the case, readers are directed to ignore the
other two.
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CHAPTER 10
APPLICATIONS in OPERATIONS MANAGEMENT
Inventory Management
© Comstock Images/Jupiterimages
Operations managers use inventory models to determine the stock level that minimizes total costs. In Section 8.2, we showed how the probabilistic model is used
to make the inventory level decision (see page 287). One component of that
model is the mean demand during lead time. Recall that lead time refers to the
interval between the time an order is made and when it is delivered. Demand
during lead time is a random variable that is often assumed to be normally distributed. There are several ways to determine mean demand during lead time, but the simplest is to estimate that quantity from a sample.
EXAMPLE
10.1
DATA
Xm10-01
Doll Computer Company
The Doll Computer Company makes its own computers and delivers them directly to
customers who order them via the Internet. Doll competes primarily on price and speed
of delivery. To achieve its objective of speed, Doll makes each of its five most popular
computers and transports them to warehouses across the country. The computers are
stored in the warehouses from which it generally takes 1 day to deliver a computer to
the customer. This strategy requires high levels of inventory that add considerably to
the cost. To lower these costs, the operations manager wants to use an inventory model.
He notes that both daily demand and lead time are random variables. He concludes that
demand during lead time is normally distributed, and he needs to know the mean to
compute the optimum inventory level. He observes 25 lead time periods and records
the demand during each period. These data are listed here. The manager would like a
95% confidence interval estimate of the mean demand during lead time. From long
experience, the manager knows that the standard deviation is 75 computers.
Demand During Lead Time
235
421
394
261
386
374
361
439
374
316
309
514
348
302
296
499
462
344
466
332
253
369
330
535
334
SOLUTION
IDENTIFY
To ultimately determine the optimum inventory level, the manager must know the
mean demand during lead time. Thus, the parameter to be estimated is ␮. At this point,
we have described only one estimator. Thus, the confidence interval estimator that we
intend to use is
x ; za>2
s
2n
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343
The next step is to perform the calculations. As we discussed previously, we will
perform the calculations in three ways: manually, using Excel, and using Minitab.
COMPUTE
MANUALLY
We need four values to construct the confidence interval estimate of ␮. They are
x, za>2, s, n
Using a calculator, we determine the summation a xi = 9,254. From this, we find
x =
9,254
a xi
=
= 370.16
n
25
The confidence level is set at 95%; thus, 1 ⫺ ␣ ⫽ .95, ␣ ⫽ 1 ⫺ .95 ⫽ .05, and
␣/2 ⫽ .025.
From Table 3 in Appendix B or from Table 10.1, we find
za>2 = z.025 = 1.96
The population standard deviation is ␴ ⫽ 75, and the sample size is 25. Substituting x,
Z␣/2, ␴, and n into the confidence interval estimator, we find
x ; za>2
s
2n
= 370.16 ; z.025
75
225
= 370.16 ; 1.96
75
225
= 370.16 ; 29.40
The lower and upper confidence limits are LCL ⫽ 340.76 and UCL ⫽ 399.56,
respectively.
EXCEL
1
2
3
4
5
6
7
8
9
A
B
z-Estimate: Mean
Mean
Standard Deviation
Observations
SIGMA
LCL
UCL
C
Demand
370.16
80.783
25
75
340.76
399.56
INSTRUCTIONS
1. Type or import the data into one column. (Open Xm10-01.)
2. Click Add-Ins, Data Analysis Plus, and Z-Estimate: Mean.
3. Fill in the dialog box: Input Range (A1:A26), type the value for the Standard
Deviation (75), click Labels if the first row contains the name of the variable, and
specify the confidence level by typing the value of ␣ (.05).
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©Vicki Beaver
DO -IT-YOURSELF
There is another way to produce the interval estimate
for this problem. If you have already calculated the
sample mean and know the sample size and
population standard deviation, you need not employ
1
2
3
4
5
6
A
z-Estimate of a Mean
Sample mean
Population standard deviation
Sample size
Confidence level
E XC E L
the data set and Data Analysis Plus described above.
Instead you can create a spreadsheet that performs
the calculations. Our suggested spreadsheet is shown
next.
B
370.16
75
25
0.95
C
Confidence Interval Estimate
370.16
Lower confidence limit
Upper confidence limit
D
E
±
29.40
340.76
399.56
Here are the tools (Excel functions) you will need to create this spreadsheet.
SQRT: Syntax: SQRT(X) Computes the square root of the quantity in parentheses.
Use the Inser t and ⍀ Symbol to input the ⫾ sign.
NORMSINV: Syntax: NORMSINV (Probability) This function calculates the value of
z such that P1Z 6 z2 ⫽ the probability in parentheses. For example, NORMSINV(.95)
determines the value of z.05, which is 1.645. You will need to figure out how to convert
the confidence level specified in Cell B6 into the value for z␣/2.
We recommend that you save the spreadsheet. It can be used to solve some of the
exercises at the end of this section.
In addition to providing another method of using Excel, this spreadsheet allows you
to perform a “what-if” analysis; that is, this worksheet provides you the opportunity to
learn how changing some of the inputs affects the estimate. For example, type 0.99 in cell
B6 to see what happens to the size of the interval when you increase the confidence level.
Type 1000 in cell B5 to examine the effect of increasing the sample size. Type 10 in cell
B4 to see what happens when the population standard deviation is smaller.
MINITAB
One-Sample Z : Demand
The assumed standard deviation = 75
Variable
Demand
N
25
Mean
370.160
StDev SE Mean
80.783
15.000
95% CI
(340.761, 399.559)
The output includes the sample standard deviation (StDev ⫽ 80.783), which is not needed
for this interval estimate. Also printed is the standard error 1SE Mean = s> 1n = 15.02,
and last, but not least, the 95% confidence interval estimate of the population mean.
INSTRUCTIONS
1. Type or import that data into one column. (Open Xm10-01.)
2. Click Stat, Basic Statistics, and 1-Sample Z . . . .
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3. Type or use the Select button to specify the name of the variable or the column it is
stored in. In the Samples in columns box (Demand), type the value of the population
standard deviation (75), and click Options . . . .
4. Type the value for the confidence level (.95) and in the Alternative box select not
equal.
INTERPRET
The operations manager estimates that the mean demand during lead time lies between
340.76 and 399.56. He can use this estimate as an input in developing an inventory policy.
The model discussed in Section 8.2 computes the reorder point, assuming a particular
value of the mean demand during lead time. In this example, he could have used the sample mean as a point estimator of the mean demand, from which the inventory policy could
be determined. However, the use of the confidence interval estimator allows the manager
to use both the lower and upper limits so that he can understand the possible outcomes.
Interpreting the Confidence Interval Estimate
Some people erroneously interpret the confidence interval estimate in Example 10.1 to
mean that there is a 95% probability that the population mean lies between 340.76 and
399.56. This interpretation is wrong because it implies that the population mean is a
variable about which we can make probability statements. In fact, the population mean
is a fixed but unknown quantity. Consequently, we cannot interpret the confidence
interval estimate of ␮ as a probability statement about ␮. To translate the confidence
interval estimate properly, we must remember that the confidence interval estimator
was derived from the sampling distribution of the sample mean. In Section 9.1, we used
the sampling distribution to make probability statements about the sample mean.
Although the form has changed, the confidence interval estimator is also a probability
statement about the sample mean. It states that there is 1 ⫺ ␣ probability that the sample mean will be equal to a value such that the interval x - za>2s> 1n to x + za>2s> 1n
will include the population mean. Once the sample mean is computed, the interval acts
as the lower and upper limits of the interval estimate of the population mean.
As an illustration, suppose we want to estimate the mean value of the distribution
resulting from the throw of a fair die. Because we know the distribution, we also know
that ␮ ⫽ 3.5 and ␴ ⫽ 1.71. Pretend now that we know only that ␴ ⫽ 1.71, that ␮ is
unknown, and that we want to estimate its value. To estimate ␮, we draw a sample of
size n ⫽ 100 and calculate x. The confidence interval estimator of ␮ is
x ; za>2
s
1n
The 90% confidence interval estimator is
x ; za>2
1.71
s
= x ; 1.645
= x ; .281
1n
2100
This notation means that, if we repeatedly draw samples of size 100 from this population, 90% of the values of x will be such that ␮ would lie somewhere between x - .281
and x + .281, and 10% of the values of x will produce intervals that would not include ␮.
Now, imagine that we draw 40 samples of 100 observations each. The values of x and the
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resulting confidence interval estimates of ␮ are shown in Table 10.2. Notice that not all
the intervals include the true value of the parameter. Samples 5, 16, 22, and 34 produce
values of x that in turn produce intervals that exclude ␮.
Students often react to this situation by asking, What went wrong with samples
5, 16, 22, and 34? The answer is nothing. Statistics does not promise 100% certainty. In
fact, in this illustration, we expected 90% of the intervals to include ␮ and 10% to
exclude ␮. Since we produced 40 intervals, we expected that 4.0 (10% of 40) intervals
would not contain ␮ ⫽ 3.5.* It is important to understand that, even when the statistics
practitioner performs experiments properly, a certain proportion (in this example, 10%)
of the experiments will produce incorrect estimates by random chance.
TABLE
10. 2
90% Confidence Interval Estimates of ␮
SAMPLE
x–
LCL ⴝ x– ⴚ .281
UCL ⴝ x– ⴙ .281
DOES INTERVAL
INCLUDE ␮ ⴝ 3.5?
1
3.550
3.269
3.831
Yes
2
3.610
3.329
3.891
Yes
3
3.470
3.189
3.751
Yes
4
3.480
3.199
3.761
Yes
5
3.800
3.519
4.081
No
6
3.370
3.089
3.651
Yes
7
3.480
3.199
3.761
Yes
8
3.520
3.239
3.801
Yes
9
3.740
3.459
4.021
Yes
10
3.510
3.229
3.791
Yes
11
3.230
2.949
3.511
Yes
12
3.450
3.169
3.731
Yes
13
3.570
3.289
3.851
Yes
14
3.770
3.489
4.051
Yes
15
3.310
3.029
3.591
Yes
16
3.100
2.819
3.381
No
17
3.500
3.219
3.781
Yes
18
3.550
3.269
3.831
Yes
19
3.650
3.369
3.931
Yes
20
3.280
2.999
3.561
Yes
21
3.400
3.119
3.681
Yes
(Continued)
*In this illustration, exactly 10% of the sample means produced interval estimates that excluded the value
of ␮, but this will not always be the case. Remember, we expect 10% of the sample means in the long run
to result in intervals excluding ␮. This group of 40 sample means does not constitute “the long run.”
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TABLE
10. 2
347
(Continued)
SAMPLE
x–
LCL ⴝ x– ⴚ .281
UCL ⴝ x– ⴙ .281
DOES INTERVAL
INCLUDE ␮ ⴝ 3.5?
22
3.880
3.599
4.161
No
23
3.760
3.479
4.041
Yes
24
3.400
3.119
3.681
Yes
25
3.340
3.059
3.621
Yes
26
3.650
3.369
3.931
Yes
27
3.450
3.169
3.731
Yes
28
3.470
3.189
3.751
Yes
29
3.580
3.299
3.861
Yes
30
3.360
3.079
3.641
Yes
31
3.710
3.429
3.991
Yes
32
3.510
3.229
3.791
Yes
33
3.420
3.139
3.701
Yes
34
3.110
2.829
3.391
No
35
3.290
3.009
3.571
Yes
36
3.640
3.359
3.921
Yes
37
3.390
3.109
3.671
Yes
38
3.750
3.469
4.031
Yes
39
3.260
2.979
3.541
Yes
40
3.540
3.259
3.821
Yes
We can improve the confidence associated with the interval estimate. If we let the
confidence level 1 ⫺ ␣ equal .95, the 95% confidence interval estimator is
x ; za>2
s
2n
= x ; 1.96
1.71
2100
= x ; .335
Because this interval is wider, it is more likely to include the value of ␮. If you redo
Table 10.2, this time using a 95% confidence interval estimator, only samples 16, 22,
and 34 will produce intervals that do not include ␮. (Notice that we expected 5% of the
intervals to exclude ␮ and that we actually observed 3/40 ⫽ 7.5%.) The 99% confidence interval estimator is
x ; za>2
s
2n
= x ; 2.575
1.71
2100
= x ; .440
Applying this interval estimate to the sample means listed in Table 10.2 would
result in having all 40 interval estimates include the population mean ␮ ⫽ 3.5. (We
expected 1% of the intervals to exclude ␮; we observed 0/40 ⫽ 0%.)
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CHAPTER 10
S E E I N G S TAT I S T I C S
applet 15 Confidence Interval Estimates of a Mean
The simulations used in the applets
line. If the confidence interval includes the
introduced in Chapter 9 can be used
true population mean of 3.5 (i.e., if the
here to demonstrate how confidence
confidence interval line overlaps the green
interval estimates are interpreted. This
vertical line), it is displayed in blue. If the
applet generates samples of size 100
confidence interval does not include the
from the population of the toss of a die.
true mean, it is displayed in red.
We know that the population mean is
After you understand the basics,
␮ ⫽ 3.5 and that the standard deviation
click the Sample 10 button a few times
is 1.71. The 95% confidence interval
to see 10 confidence intervals (but not
estimator is
their calculations) at once. Then click on
x ; za>2
s
2n
= x ; 1.96
1.71
2100
= x ; .335
The applet will generate one sample,
10 samples, or 100 samples at a time. The
resulting confidence interval is displayed
as a horizontal line between the upper
the Sample 100 button to generate
100 samples and confidence intervals.
Applet Exercises
15.3 How many intervals did you expect
Simulate 100 samples.
to see that correctly included the
15.1 Are all the confidence interval
mean?
estimates identical?
15.2 Count the number of confidence
15.4 What do these exercises tell
you about the proper interpre-
and lower ends of the confidence interval.
interval estimates that include the
tation of a confidence interval
The true mean of 3.5 is the green vertical
true value of the mean.
estimate?
In actual practice, only one sample will be drawn, and thus only one value of x
will be calculated. The resulting interval estimate will either correctly include the
parameter or incorrectly exclude it. Unfortunately, statistics practitioners do not
know whether they are correct in each case; they know only that, in the long run, they
will incorrectly estimate the parameter some of the time. Statistics practitioners
accept that as a fact of life.
We summarize our calculations in Example 10.1 as follows. We estimate that the
mean demand during lead time falls between 340.76 and 399.56, and this type of estimator is correct 95% of the time. Thus, the confidence level applies to our estimation procedure and not to any one interval. Incidentally, the media often refer to the 95% figure
as “19 times out of 20,” which emphasizes the long-run aspect of the confidence level.
Information and the Width of the Interval
Interval estimation, like all other statistical techniques, is designed to convert data into
information. However, a wide interval provides little information. For example, suppose that as a result of a statistical study we estimate with 95% confidence that the average starting salary of an accountant lies between $15,000 and $100,000. This interval is
so wide that very little information was derived from the data. Suppose, however, that
the interval estimate was $52,000 to $55,000. This interval is much narrower, providing
accounting students more precise information about the mean starting salary.
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349
The width of the confidence interval estimate is a function of the population standard deviation, the confidence level, and the sample size. Consider Example 10.1,
where ␴ was assumed to be 75. The interval estimate was 370.16 ⫾ 29.40. If ␴ equaled
150, the 95% confidence interval estimate would become
x ; za>2
s
2n
= 370.16 ; z.025
150
225
= 370.16 ; 1.96
150
225
= 370.16 ; 58.80
Thus, doubling the population standard deviation has the effect of doubling the width
of the confidence interval estimate. This result is quite logical. If there is a great deal of
variation in the random variable (measured by a large standard deviation), it is more difficult to accurately estimate the population mean. That difficulty is translated into a
wider interval.
Although we have no control over the value of ␴, we do have the power to select
values for the other two elements. In Example 10.1, we chose a 95% confidence level. If
we had chosen 90% instead, the interval estimate would have been
x ; za>2
s
2n
= 370.16 ; z.05
75
225
= 370.16 ; 1.645
75
225
= 370.16 ; 24.68
A 99% confidence level results in this interval estimate:
x ; za>2
s
2n
= 370.16 ; z.005
75
225
= 370.16 ; 2.575
75
225
= 370.16 ; 38.63
As you can see, decreasing the confidence level narrows the interval; increasing it
widens the interval. However, a large confidence level is generally desirable because
that means a larger proportion of confidence interval estimates that will be correct in
the long run. There is a direct relationship between the width of the interval and the
confidence level. This is because we need to widen the interval to be more confident in
the estimate. (The analogy is that to be more likely to capture a butterfly, we need a
larger butterfly net.) The trade-off between increased confidence and the resulting
wider confidence interval estimates must be resolved by the statistics practitioner. As a
general rule, however, 95% confidence is considered “standard.”
The third element is the sample size. Had the sample size been 100 instead of 25,
the confidence interval estimate would become
x ; za>2
s
2n
= 370.16 ; z.025
75
2100
= 370.16 ; 1.96
75
2100
= 370.16 ; 14.70
Increasing the sample size fourfold decreases the width of the interval by ha